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Starshaped sets. (English) Zbl 1454.52001

A subset \(S\) of \(\mathbb{R}^d\) (say) is called starshaped if there exists a point \(x\in S\) such that for each \(y\in S\) the segment with endpoints \(x\) and \(y\) belongs to \(S\). In the words of the authors, this is an expository paper “emphasizing the geometric, analytical, combinatorial, and topological properties of starshaped sets and their broad applicability in many mathematical fields”. The authors have meticulously combed the existent literature and have documented and described a huge number of appearances of starshaped sets (563 references). It seems safe to predict that every interested reader will find something that is new for him/her and of value. The first part of the survey deals with structural properties of starshaped sets, relations between starshaped sets, variants and generalizations. The wealth of the presented material can here only be illuminated by quoting the section headings. 1. Introduction, 2. Basic notions and definitions, 3. Cones, 4. Starshaped sets and visibility, 5. Star generators. Representations of the kernel, 6. Krasnosel’skii-type theorems, 7. Asymptotic structure of starshaped sets, 8. Support cones, 9. Separation of starshaped sets, 10. Extremal structure of starshaped sets, 11. Dimension of the kernel of a starshaped set, 12. Admissible kernels of starshaped sets, 13. Radial functions of starshaped sets, 14. Sums, unions, and intersections of starshaped sets, 15. Spaces of starshaped sets, 16. Selectors for star bodies, 17. Star duality, intersection bodies, and related topics, 18. Extensions and generalizations.
Already in the first part, instances are mentioned where generalizations of classical theories from convexity necessitate the use of starshaped sets and radial functions. So Sections 14 and 17 discuss, for example, the dual Brunn-Minkowski inequality, the Orlicz-Brunn-Minkowski theory and its dual, intersection bodies, valuations on star bodies, and several other topics. The second part of this survey gives a broad picture of the use of starshaped sets in various fields from pure and applied mathematics. A quotation of the section headings may give a picture of the diversity of these applications. 19.1. Discrete and computational geometry, 19.2. Inequalities, 19.3. Starshapedness in differential geometry, 19.4. Starshaped sets and PDE, 19.5. Starshapedness in fixed point theory, 19.6. Starshaped sets in approximation theory, 19.7. Applications of starshapedness in optimization. This overwhelming picture of the numerous appearances of starshaped sets in various different surroundings concludes with 19.8. Further topics. This section considers, for example, fractal star bodies, starshaped Kakeya sets, random starshaped sets, typical starshaped sets in the Baire category sense. We mentioned these here to emphasize the extremely wide range of the topic of starshaped sets.

MSC:

52-02 Research exposition (monographs, survey articles) pertaining to convex and discrete geometry
52A30 Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.)
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[1] Abbas, M.; Rhoades, BE, A fixed point result for asymptotically nonexpansive mappings on an unbounded set, Carpath. J. Math., 25, 2, 141-146 (2009) · Zbl 1249.47044
[2] Aeppli, A., On the uniqueness of compact solutions for certain elliptic differential equations, Proc. Am. Math. Soc., 11, 826-832 (1960) · Zbl 0196.40502
[3] Akashi, S.; Takahashi, W., Strong convergence theorem for nonexpansive mappings on star-shaped sets in Hilbert spaces, Appl. Math. Comput., 219, 4, 2035-2040 (2012) · Zbl 1332.47036
[4] Akdoğan, S., A theorem for locally starshaped sets, Rend. Mat. Appl. (7), 10, 2, 201-204 (1990) · Zbl 0724.52004
[5] Akkouchi, M., A contraction principle in weakly Cauchy normed spaces, Nonlinear Funct. Anal. Appl., 15, 3, 481-486 (2010) · Zbl 1244.54084
[6] Alexander, J., Functions which map the interior of the unit circle upon a simple region, Ann. Math. (2), 17, 1, 12-22 (1915) · JFM 45.0672.02
[7] Alexander, R., Edelstein, M.: Finite visibility and starshape in Hilbert space. Preprint
[8] Alfonseca, MA; Cordier, M.; Ryabogin, D., On bodies with directly congruent projections and sections, Isr. J. Math., 215, 2, 765-799 (2016) · Zbl 1358.52007
[9] Alonso-Gutiérrez, D.; Henk, M.; Hernández Cifre, MA, A characterization of dual quermassintegrals and the roots of dual Steiner polynomials, Adv. Math., 331, 565-588 (2018) · Zbl 1393.52005
[10] Al-Shamary, B.; Mishra, SK; Laha, V., On approximate starshapedness in multiobjective optimization, Optim. Methods Softw., 31, 2, 290-304 (2016) · Zbl 1360.90209
[11] Al-Thagafi, MA, Common fixed points and best approximation, J. Approx. Theory, 85, 3, 318-323 (1996) · Zbl 0858.41022
[12] Amir, D.; Lindenstrauss, J., The structure of weakly compact sets in Banach spaces, Ann. Math. (2), 88, 35-46 (1968) · Zbl 0164.14903
[13] Asplund, E., A \(k\)-extreme point is the limit of \(k\)-exposed points, Isr. J. Math., 1, 161-162 (1963) · Zbl 0125.11201
[14] Aurenhammer, F.; Klein, R.; Lee, D-T, Voronoi Diagrams and Delaunay Triangulations (2013), Hackensack: World Scientific, Hackensack · Zbl 1295.52001
[15] Aussel, D.; Ye, JJ, Quasiconvex programming with locally starshaped constraint region and applications to quasiconvex MPEC, Optimization, 55, 5-6, 433-457 (2006) · Zbl 1134.49010
[16] Azagra, D.; Cepedello Boiso, M., Smooth Lipschitz retractions of starlike bodies onto their boundaries in infinite-dimensional Banach spaces, Bull. Lond. Math. Soc., 33, 4, 443-453 (2001) · Zbl 1032.46097
[17] Azagra, D.; Deville, R., James’ theorem fails for starlike bodies, J. Funct. Anal., 180, 2, 328-346 (2001) · Zbl 0983.46016
[18] Azagra, D.; Dobrowolski, T., On the topological classification of starlike bodies in Banach spaces, Topol. Appl., 132, 3, 221-234 (2003) · Zbl 1031.52001
[19] Azagra, D.; Montesinos, A., Starlike bodies and deleting diffeomorphisms in Banach spaces, Extracta Math., 19, 2, 171-213 (2004) · Zbl 1079.46051
[20] Baildon, J.D.: Finitely starlike sets and refinements of Helly’s theorem. In: Proceedings of the 16th Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1985). Congr. Numer. 49 (1985), pp. 5-10 · Zbl 0629.52009
[21] Baildon, JD; Silverman, R., On starshaped sets and Helly-type theorems, Pac. J. Math., 62, 1, 37-41 (1976) · Zbl 0329.52008
[22] Baildon, J.D., Silverman, R.: Combinatorial properties of Helly-type sets. In: Proceedings of the 9th Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, FL, 1978), pp. 77-84, Congress. Numer., XXI, Utilitas Math., Winnipeg, Man (1978) · Zbl 0415.52006
[23] Baillo, A.; Cuevas, A., On the estimation of a star-shaped set, Adv. Appl. Prob., 33, 717-726 (2001) · Zbl 1003.62030
[24] Bair, J.; Jongmans, F., Sur l’énigme de l’enveloppe conique fermé, Bull. Soc. Roy. Sc. Liège, 52, 285-294 (1983) · Zbl 0526.52001
[25] Bakelman, I.J., Kantor, B.E.: Existence of a hypersurface homeomorphic to the sphere in Euclidean space with a given mean curvature. Geometry and Topology, No. 1 (Russian), pp. 3-10. Leningrad. Gos. Ped. Inst. im. Gercena, Leningrad (1974) (in Russian)
[26] Bambah, RP, On the geometry of numbers of non-convex star-regions with hexagonal symmetry, Philos. Trans. R. Soc. Lond. Ser. A., 243, 431-462 (1951) · Zbl 0044.27102
[27] Bárány, I.; Matoušek, J., Berge’s theorem, fractional Helly, and art galleries, Discrete Math., 306, 19-20, 2303-2313 (2006) · Zbl 1103.52003
[28] Barbosa, JLM; de Lira, JHS; Oliker, V., Uniqueness of starshaped compact hypersurfaces with prescribed \(m\)-th mean curvature in hyperbolic space, Ill. J. Math., 51, 2, 571-582 (2007) · Zbl 1128.53007
[29] Baronti, M.; Casini, E.; Papini, PL, Nested sequences of stars and starshaped sets, J. Math. Anal. Appl., 477, 1, 685-691 (2019) · Zbl 1429.46010
[30] Barthel, W.; Pabel, H., Das isodiametrische Problem der Minkowski-Geometrie, Results Math., 12, 3-4, 252-267 (1987) · Zbl 0633.52010
[31] Beer, GA, The continuity of the visibility function on a starshaped set, Can. J. Math., 24, 989-992 (1972) · Zbl 0229.52007
[32] Beer, GA, The index of convexity and the visibility function, Pac. J. Math., 44, 59-67 (1973) · Zbl 0262.52005
[33] Beer, GA, Starshaped sets and the Hausdorff metric, Pac. J. Math., 61, 21-27 (1975) · Zbl 0327.52004
[34] Beer, GA, On closed starshaped sets and Baire category, Proc. Am. Math. Soc., 78, 555-558 (1980) · Zbl 0446.52003
[35] Beer, GA; Klee, VL, Limits of starshaped sets, Arch. Math., 48, 241-249 (1987) · Zbl 0603.46014
[36] Beer, G.; Villar, L., On the approximation of starshaped sets in Hausdorff distance, Serdica, 13, 4, 403-407 (1987) · Zbl 0694.41042
[37] Beg, I.; Abbas, M., Random fixed points of asymptotically nonexpansive random operators on unbounded domains, Math. Slovaca, 58, 6, 755-762 (2008) · Zbl 1199.47236
[38] Beg, I.; Azam, A., Fixed points on star-shaped subsets of convex metric spaces, Indian J. Pure Appl. Math., 18, 7, 594-596 (1987) · Zbl 0622.54033
[39] Beltagy, M., On starshaped sets, Bull. Malays. Math. Soc. (2), 11, 2, 49-57 (1988) · Zbl 0689.53006
[40] Beltagy, M., A comparison study of convex and starshaped subsets, Delta J. Sci., 13, 3, 1179-1190 (1989)
[41] Beltagy, M., Immersions into manifolds without conjugate points, J. Inst. Math. Comput. Sci. Math., Ser., 3, 3, 265-271 (1990) · Zbl 0844.53043
[42] Beltagy, M.: Convex and starshaped subsets in manifolds product. Comm. Fac. Sci. Univ. Ankara Ser. \(A_1\) Math. Statist. 41(12), 35-44 (1992-1994) · Zbl 0829.53036
[43] Beltagy, M., Conditional imbedding into manifolds without conjugate points, Bull. Calcutta Math. Soc., 87, 2, 119-122 (1995) · Zbl 0829.53048
[44] Beltagy, M.; El-Araby, A., On convex and starshaped hulls, Kyungpook Math. J., 40, 2, 313-321 (2000) · Zbl 0986.52006
[45] Beltagy, M.; El-Araby, A., Starshaped sets in Riemannian manifolds without conjugate points, Far East J. Math. Sci. (FJMS), 6, 2, 187-196 (2002) · Zbl 1036.53020
[46] Beltagy, M.; Shenawy, S., Sets with zero-dimensional kernels, Int. J. Mod. Math., 4, 2, 163-168 (2009) · Zbl 1175.52004
[47] Beltagy, M.; Shenawy, S., A note on convexity and starshapedness, Appl. Math. Sci. (Ruse), 4, 53-56, 2599-2608 (2010) · Zbl 1234.52005
[48] Ben-El-Mechaiekh, H., The Ky Fan fixed point theorem on star-shaped domains, C. R. Math. Acad. Sci. Soc. R. Can., 27, 4, 97-100 (2005) · Zbl 1094.47048
[49] Berck, G., Convexity of \(L_p\)-intersection bodies, Adv. Math., 222, 3, 920-936 (2009) · Zbl 1179.52005
[50] Berestycki, H.; Hamel, F.; Matano, H., Bistable traveling waves around an obstacle, Commun. Pure Appl. Math., 62, 6, 729-788 (2009) · Zbl 1172.35031
[51] Berestycki, H.; Lasry, J-M; Mancini, G.; Ruf, B., Existence of multiple periodic orbits on star-shaped Hamiltonian surfaces, Commun. Pure Appl. Math., 38, 3, 253-289 (1985) · Zbl 0569.58027
[52] Bezdek, K.; Naszódi, M., Spindle starshaped sets, Aequ. Math., 89, 3, 803-819 (2015) · Zbl 1331.52011
[53] Bobylev, NA, The Helly theorem for star-shaped sets. Pontryagin Conference, 8, Topology (Moscow, 1998), J. Math. Sci. (New York), 105, 2, 1819-1825 (2001) · Zbl 1013.52006
[54] Bobylev, NA, Some remarks on star-shaped sets, Mat. Zametki, 65, 4, 511-519 (1999) · Zbl 0980.52004
[55] Böröczky, K.J., Schneider, R.: Stable determination of convex bodies from sections. Stud. Sci. Math. Hung. 46(3), 367-376 (2009) · Zbl 1224.52008
[56] Böttcher, R.; Hecker, H-D, Streckensternförmigkeit—eine weitere Verallgemeinerung der Sternförmigkeit, Beitr. Algebra Geom., 33, 109-114 (1992) · Zbl 0763.52003
[57] Bollobás, G., Star domains, Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 9, 67-70 (1966) · Zbl 0149.39904
[58] Boltyanski, V.; Martini, H.; Soltan, PS, Star-shaped sets in normed spaces, Discrete Comput. Geom., 15, 1, 63-71 (1996) · Zbl 0847.52003
[59] Boltyanski, V., Martini, H., Soltan, P.S.: Excursions into Combinatorial Geometry. Universitext, Springer, Berlin et al. (1997) · Zbl 0877.52001
[60] Boltyanski, V., Soltan, P.S.: Star-shaped sets (Russian). Bul. Akad. Štiince RSS Moldoven. 3, 7-11, 92 (1976) · Zbl 0335.52006
[61] Boltyanski, V., Soltan, P.S.: Combinatorial Geometry of Various Classes of Convex Sets (Russian). “Shtiinca”, Kishinev (1978) · Zbl 0528.52002
[62] Borwein, JM, A proof of the equivalence of Helly’s and Krasnosel’skii’s theorem, Can. Math. Bull., 20, 35-37 (1977) · Zbl 0375.52008
[63] Borwein, JM, Tangent cones, starshape and convexity, Int. J. Math. Math. Sci., 1, 4, 459-477 (1978) · Zbl 0438.52009
[64] Borwein, JM, Completeness and the contraction principle, Proc. Am. Math. Soc., 87, 2, 246-250 (1983) · Zbl 0511.47039
[65] Borwein, JM; Edelstein, M.; O’Brien, R., Visibility and starshape, J. Lond. Math. Soc. (2), 14, 313-318 (1976) · Zbl 0343.46012
[66] Borwein, JM; Edelstein, M.; O’Brien, R., Some remarks on visibility and starshape, J. Lond. Math. Soc. (2), 15, 2, 342-344 (1977) · Zbl 0357.46021
[67] Borwein, JM; Lewis, A., Convex Analysis and Nonlinear Optimization (2000), Berlin: Springer, Berlin · Zbl 0953.90001
[68] Bragard, L., Ensembles étoilés et irradiés de \(\mathbb{R}^n \), Bull. Soc. R. Sc. Liège, 36, 238-243 (1967) · Zbl 0148.42701
[69] Bragard, L., Ensembles étoilés et irradiés dans un espace vectoriel topologique, Bull. Soc. R. Sc. Liège, 37, 274-285 (1968) · Zbl 0172.39305
[70] Bragard, L., Ensembles irradiés et composantes convexes, Bull. Soc. R. Sc. Liège, 38, 649-653 (1969) · Zbl 0192.46801
[71] Bragard, L., Décomposition d’un ensemble étoilé, Bull. Soc. R. Sc. Liège, 39, 114-117 (1970) · Zbl 0195.12602
[72] Bragard, L., Décomposition des ensembles irradiés, Bull. Soc. R. Sc. Liège, 39, 264-268 (1970) · Zbl 0199.43701
[73] Bragard, L., Charactérisation du mirador d’un ensemble dans un espace vectoriel, Bull. Soc. R. Sc. Liège, 39, 260-263 (1970) · Zbl 0199.43604
[74] Bragard, L., Cônes étoilés et cônes asymptotes, Bull. Soc. R. Sc. Liège, 41, 20-23 (1972) · Zbl 0233.46014
[75] Bragard, L., Cônes visuels, composantes convexes et ensembles étoilés, Bull. Soc. R. Sc. Liège, 41, 640-651 (1972) · Zbl 0256.52007
[76] Bragard, L., Cônes associés à un ensemble, Bull. Soc. R. Sc. Liège, 42, 549-560 (1973) · Zbl 0279.46006
[77] Breen, M., Sets in \(\mathbb{R}^d\) having \((d-2)\)-dimensional kernels, Pac. J. Math., 75, 1, 37-44 (1978) · Zbl 0369.52011
[78] Breen, M., Sets with \((d-2)\)-dimensional kernels, Pac. J. Math., 77, 1, 51-55 (1978) · Zbl 0386.52007
[79] Breen, M., A Helly type theorem for the dimension of the kernel of starshaped set, Proc. Am. Math. Soc., 73, 233-236 (1979) · Zbl 0371.52002
[80] Breen, M., The dimension of the kernel of a planar set, Pac. J. Math., 82, 15-21 (1979) · Zbl 0377.52008
[81] Breen, M., \((d-2)\)-extreme subsets and a Helly-type theorem for starshaped sets, Can. J. Math., 32, 3, 707-713 (1980) · Zbl 0411.52005
[82] Breen, M., A quantitative version of Krasnosel’skii’s theorem in \(\mathbb{R}^2 \), Pac. J. Math., 91, 1, 31-37 (1980) · Zbl 0423.52006
[83] Breen, M., \(k\)-dimensional intersections of convex sets and convex kernels, Discrete Math., 36, 233-237 (1981) · Zbl 0464.52005
[84] Breen, M., Admissible kernels for starshaped sets, Proc. Am. Math. Soc., 82, 622-628 (1981) · Zbl 0476.52011
[85] Breen, M., Clear visibility and the dimension of kernels of starshaped sets, Proc. Am. Math. Soc., 85, 414-418 (1982) · Zbl 0496.52007
[86] Breen, M., Points of local nonconvexity and sets which are almost starshaped, Geom. Dedic., 13, 201-213 (1982) · Zbl 0511.52004
[87] Breen, M., A Krasnosels’skii-type theorem for points of local nonconvexity, Proc. Am. Math. Soc., 85, 261-266 (1982) · Zbl 0496.52006
[88] Breen, M., A quantitative Krasnosel’skii’s theorem in \(\mathbb{R}^d \), Geom. Dedic., 12, 2, 219-226 (1982) · Zbl 0481.52007
[89] Breen, M., A Krasnosel’skii-type theorem for nonclosed sets in the plane, J. Geom., 18, 1, 28-42 (1982) · Zbl 0503.52007
[90] Breen, M., Clear visibility, starshaped sets and finitely starshaped sets, J. Geom., 19, 183-196 (1982) · Zbl 0507.52007
[91] Breen, M., An improved Krasnosel’skii theorem for nonclosed sets in the plane, J. Geom., 21, 1, 97-100 (1983) · Zbl 0525.52010
[92] Breen, M., Points of local nonconvexity, clear visibility and starshaped sets in \(\mathbb{R}^d \), J. Geom., 21, 42-52 (1983) · Zbl 0525.52011
[93] Breen, M., Clear visibility and sets which are almost starshaped, Proc. Am. Math. Soc., 91, 607-610 (1984) · Zbl 0521.52007
[94] Breen, M., Clear visibility and unions of two starshaped sets in the plane, Pac. J. Math., 115, 267-275 (1984) · Zbl 0501.52008
[95] Breen, M., A Krasnosel’skii-type theorem for unions of two starshaped sets in the plane, Pac. J. Math., 120, 19-31 (1985) · Zbl 0571.52006
[96] Breen, M.: Krasnosel’skii-type theorems. In: Discrete Geometry and Convexity (New York, 1982), pp. 142-146, Ann. New York Acad. Sci., vol. 440. New York Acad. Sci., New York (1985) · Zbl 0572.52017
[97] Breen, M., Improved Krasnosel’skii theorems for the dimension of the kernel of a starshaped set, J. Geom., 27, 175-179 (1986) · Zbl 0611.52006
[98] Breen, M., A Krasnosel’skii theorem for nonclosed sets in \(\mathbb{R}^d \), J. Geom., 26, 105-114 (1986) · Zbl 0588.52009
[99] Breen, M., Determining starshaped sets and unions of starshaped sets by their sections, J. Geom., 28, 1, 80-85 (1987) · Zbl 0616.52008
[100] Breen, M., \(k\)-partitions and a characterization for compact unions of starshaped sets, Proc. Am. Math. Soc., 102, 3, 677-680 (1988) · Zbl 0644.52004
[101] Breen, M., A weak Krasnosel’skii theorem in \(\mathbb{R}^d \), Proc. Am. Math. Soc., 104, 558-562 (1988) · Zbl 0691.52007
[102] Breen, M., Characterizing compact unions of two starshaped sets in \(\mathbb{R}^d \), J. Geom., 35, 14-19 (1989) · Zbl 0678.52006
[103] Breen, M., Unions of three starshaped sets in \(\mathbb{R}^2 \), J. Geom., 36, 8-16 (1989) · Zbl 0689.52002
[104] Breen, M., Finitely starlike sets whose F-stars have positive measure, J. Geom., 35, 19-25 (1989) · Zbl 0691.52005
[105] Breen, M., Starshaped unions and nonempty intersections of convex sets in \(\mathbb{R}^d \), Proc. Am. Math. Soc., 108, 3, 817-820 (1990) · Zbl 0691.52006
[106] Breen, M., The dimension of the kernel in an intersection of starshaped sets, Arch. Math. (Basel), 81, 4, 485-490 (2003) · Zbl 1044.52004
[107] Breen, M., A Helly-type theorem for countable intersections of starshaped sets, Arch. Math., 84, 3, 282-288 (2005) · Zbl 1080.52006
[108] Breen, M., Analogues of Horn’s theorem for finite unions of starshaped sets in \(\mathbb{R}^d \), Period. Math. Hungar., 59, 1, 99-107 (2009) · Zbl 1224.52010
[109] Breen, M., Suitable families of boxes and kernels of staircase starshaped sets in \(\mathbb{R}^d \), Aequat. Math., 87, 43-52 (2014) · Zbl 1314.52007
[110] Breen, M., Intersections of sets expressible as unions of \(k\) starshaped sets, Ars Combin., 125, 339-345 (2016) · Zbl 1413.52007
[111] Breen, M.; Zamfirescu, T., A characterization theorem for certain unions of two starshaped sets in \(\mathbb{R}^2 \), Geom. Dedic., 6, 95-103 (1987) · Zbl 0611.52007
[112] Brehm, U., Convex bodies with non-convex cross-section bodies, Mathematika, 46, 1, 127-129 (1999) · Zbl 0988.52018
[113] Brendle, S.; Hung, P-K; Wang, M-T, A Minkowski inequality for hypersurfaces in the anti-de Sitter-Schwarzschild manifold, Commun. Pure Appl. Math., 69, 1, 124-144 (2016) · Zbl 1331.53078
[114] Bressan, JC, Estrellados y separabilidad en un sistema axiomático para la convexidad, Rev. Un. Mat. Argentina, 31, 1-5 (1983)
[115] Brown, JG, A note on fuzzy sets, Inform. Control, 18, 32-39 (1971) · Zbl 0217.01403
[116] Bruckner, AM; Bruckner, JB, On \(L_n\) sets, the Hausdorff metric and connectedness, Proc. Am. Math. Soc., 13, 765-767 (1962) · Zbl 0131.38005
[117] Brunn, H., Über Kerneigebiete, Math. Ann., 73, 436-440 (1913) · JFM 44.0560.01
[118] Bshouty, D.; Hengartner, N.; Hengartner, W., A constructive method for starlike harmonic mappings, Numer. Math., 54, 2, 167-178 (1988) · Zbl 0676.30014
[119] Buchman, E.; Valentine, F., External visibility, Pac. J. Math., 64, 333-340 (1972) · Zbl 0346.52008
[120] Busemann, H.; Petty, CM, Problems on convex bodies, Math. Scand., 4, 88-94 (1956) · Zbl 0070.39301
[121] Caffarelli, L., Nirenberg, L., Spruck, J.: Nonlinear second order elliptic equations. IV. Starshaped compact Weingarten hypersurfaces. In: Current Topics in Partial Differential Equations, pp. 1-26, Kinokuniya, Tokyo (1986) · Zbl 0672.35027
[122] Calini, A.; Ivey, T.; Marí-Beffa, G., Remarks on KdV-type flows on star-shaped curves, Physica D, 238, 8, 788-797 (2009) · Zbl 1218.37102
[123] Campi, S., On the reconstruction of a star-shaped body from its “half-volumes”, J. Aust. Math. Soc. Ser. A, 37, 2, 243-257 (1984) · Zbl 0563.53052
[124] Carbone, A., Extensions of a few fixed point theorems, J. Indian Acad. Math., 28, 1, 125-131 (2006) · Zbl 1131.47051
[125] Castillo, JMF; Papini, PL, Approximation of the limit distance function in Banach spaces, J. Math. Anal. Appl., 328, 1, 577-589 (2007) · Zbl 1119.46017
[126] Cel, J., Determining dimension of the kernel of a cone, Monatsh. Math., 114, 2, 83-88 (1992) · Zbl 0765.52010
[127] Cel, J., Solution of the problem of combinatorial characterization of the dimension of the kernel of a starshaped set, J. Geom., 53, 28-36 (1995) · Zbl 0831.52003
[128] Cel, J., An optimal Krasnosel’skii-type theorem for the dimension of the kernel of a starshaped set, Bull. Lond. Math. Soc., 27, 249-256 (1995) · Zbl 0823.46007
[129] Cel, J., An optimal Krasnosel’skii-type theorem for an open starshaped set, Geom. Dedic., 66, 293-301 (1997) · Zbl 0908.46005
[130] Cel, J., Sets which are almost starshaped, J. Geom., 62, 36-39 (1998) · Zbl 0926.52008
[131] Cel, J., Characterizing starshaped sets by maximal visibility, Geom. Dedic., 74, 135-137 (1999) · Zbl 0928.46004
[132] Cel, J., Representations of starshaped sets in normed linear spaces, J. Funct. Anal., 174, 264-273 (2000) · Zbl 0967.52002
[133] Chan, JB, A Krasnosel’skii-type theorem involving \(p\)-arcs, Proc. Am. Math. Soc., 102, 667-676 (1988) · Zbl 0651.52008
[134] Chandler, E.; Faulkner, G., Fixed points in nonconvex domains, Proc. Am. Math. Soc., 80, 4, 635-638 (1980) · Zbl 0447.47044
[135] Chandok, S.; Narang, TD, On common fixed points and best approximation on nonconvex sets, Thai J. Math., 7, 2, 285-292 (2009) · Zbl 1203.41017
[136] Chandrasekaran, K., Dadush, D., Vempala, S.: Thin partitions: isoperimetric inequalities and a sampling algoritm for star-shaped bodies. In: Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1630-1645. SIAM, Philadelphia (2010) · Zbl 1288.90062
[137] Chen, D.; Li, H.; Wang, Z., Starshaped compact hypersurfaces with prescribed Weingarten curvature in warped product manifolds, Calc. Var. Partial Differ. Equ., 57, 2, Art. 42, 26 (2018) · Zbl 1395.53069
[138] Chow, B.; Liou, L-P; Tsai, D-H, Expansion of embedded curves with turning angle greater than \(\pi \), Invent. Math., 123, 3, 415-429 (1996) · Zbl 0858.53001
[139] Cieślak, W., Miernowski, A., Mozgawa, W.: Isoptics of a closed strictly convex curve. In: Global Differential Geometry and Global Analysis (Berlin, 1990), pp. 28-35, Lecture Notes in Math., vol. 1481. Springer, Berlin (1991) · Zbl 0739.53001
[140] Colgen, R.: Stability for almost convex optimization problems. In: Proceedings of the 6th Sympos. Oper. Res., Part 1 (Augsburg, 1981), pp. 43-51. In: Methods Oper. Res., vol. 43. Athenäum/Hain/Hanstein, Königstein/Ts (1981) · Zbl 0516.65043
[141] Conrad, F.; Rao, B., Decay of solutions of the wave equation in a star-shaped domain with nonlinear boundary feedback, Asymptot. Anal., 7, 3, 159-177 (1993) · Zbl 0791.35011
[142] Cosner, C.; Schmitt, K., On the geometry of level sets of positive solutions of semilinear elliptic equations, Rocky Mountain J. Math., 18, 2, 277-286 (1988) · Zbl 0706.35022
[143] Coxeter, HSM, Regular Polytopes (1973), New York: Dover, New York
[144] Crasta, G.; Fragalà, I., A new symmetry criterion based on the distance function and applications to PDE’s, J. Differ. Equ., 255, 7, 2082-2099 (2013) · Zbl 1292.35175
[145] Crespi, GP; Ginchev, I.; Rocca, M., Minty variational inequalities, increase-along-rays property and optimization, J. Optim. Theory Appl., 123, 3, 479-496 (2004) · Zbl 1059.49010
[146] Crespi, GP; Ginchev, I.; Rocca, M., Existence of solutions and star-shapedness in Minty variational inequalities, J. Glob. Optim., 32, 4, 485-494 (2005) · Zbl 1097.49007
[147] Crespi, G.P., Rocca, M., Ginchev, I.: On a connection among Minty variational inequalities and generalized convexity. In: Recent Advances in Optimization (Varese, 2002), pp. 35-40. Datanova, Milan (2003) · Zbl 1063.49007
[148] Croft, H.T., Falconer, K.J., Guy, R.K.: Unsolved Problems in Geometry. Corrected reprint of the 1991 original. Problem Books in Mathematics. Unsolved Problems in Intuitive Mathematics, II. Springer, New York (1994)
[149] Cunnigham, F., The Kakeya problem for simply connected and star shaped sets, Am. Math. Monthly, 78, 114-129 (1971) · Zbl 0207.20903
[150] Danzer, L., Grünbaum, B., Klee, V.: Helly’s theorem and its relatives. In: Convexity (Ed. V. Klee), Proc. Symp. Pure Math., vol. 7, pp. 101-179. American Mathematical Society, New York (1963) · Zbl 0132.17401
[151] Day, MM, Normed Linear Spaces (1973), Berlin: Springer, Berlin · Zbl 0268.46013
[152] De Blasi, FS; Myjak, J., Ambiguous loci of the nearest point mapping in Banach spaces, Arch. Math. (Basel), 61, 4, 377-384 (1993) · Zbl 0822.46012
[153] De Blasi, FS; Myjak, J., Ambiguous loci of the farthest distance mapping from compact convex sets, Stud. Math., 112, 2, 99-107 (1995) · Zbl 0818.52002
[154] De Blasi, FS; Myjak, J.; Papini, PL, Starshaped sets and best approximation, Arch. Math. (Basel), 56, 1, 41-48 (1991) · Zbl 0747.49010
[155] De Blasi, FS; Kenderov, PS; Myjak, J., Ambiguous loci of the metric projection onto compact starshaped sets in a Banach space, Monatsh. Math., 119, 1-2, 23-36 (1995) · Zbl 0818.54026
[156] Delanoë, Ph.: Plongements radiaux \(S^n\hookrightarrow \mathbb{R}^{n+1}\) à courbure de Gauss positive prescrite (French. English summary). Ann. Sci. École Norm. Sup. (4) 18(4), 635-649 (1985) · Zbl 0594.53039
[157] Demianov, VF; Rubinov, A., Quasidifferentiability and Related Topics (2000), Berlin: Springer, Berlin
[158] Dem’yanovich, Yu.K., Chirkov, M.K.: Numerical approximation of star surfaces (Russian. English summary). Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 1991, vyp. 1, pp. 20-24; translation in: Vestnik Leningrad Univ. Math., 24(1), 24-29 (1991) · Zbl 0735.65005
[159] Deneen, L.; Shute, G., Polygonizations of point sets in the plane, Discrete Comput. Geom., 3, 1, 77-87 (1988) · Zbl 0629.52008
[160] Diamond, P., A note on fuzzy starshaped fuzzy sets, Fuzzy Sets Syst., 37, 2, 193-199 (1990) · Zbl 0702.54008
[161] Diamond, P.; Kloeden, P., A note on compact sets in spaces of subsets, Bull. Aust. Math. Soc., 38, 3, 393-395 (1988) · Zbl 0643.54013
[162] Diamond, P.; Kloeden, PE, Metric Spaces of Fuzzy Sets: Theory and Applications (1994), Singapore: World Scientific, Singapore · Zbl 0873.54019
[163] Diaz, J.I., Kawohl, B.: Convexity and starshapedness of level sets for some nonlinear parabolic problems. Free Boundary Problems: Theory and Applications, Vol. II (Irsee, 1987), pp. 883-887, Pitman Res. Notes Math. Ser., vol. 186. Longman Sci. Tech., Harlow (1990)
[164] Diaz, JI; Kawohl, B., On convexity and starshapedness of level sets for some nonlinear elliptic and parabolic problems on convex rings, J. Math. Anal. Appl., 177, 1, 263-286 (1993) · Zbl 0802.35087
[165] Diestel, J., Geometry of Banach Spaces-Selected Topics (1975), Berlin: Springer, Berlin · Zbl 0307.46009
[166] Ding, Q., The inverse mean curvature flow in rotationally symmetric spaces, Chin. Ann. Math. Ser. B, 32, 1, 27-44 (2011) · Zbl 1211.53084
[167] Dobkin, DP; Edelsbrunner, H.; Overmars, MH, Searching for empty convex polygons, Algorithmica, 5, 4, 561-571 (1990) · Zbl 0697.68034
[168] Dotson, W.G. Jr.: Fixed point theorems for non-expansive mappings on star-shaped subsets of Banach spaces. J. Lond. Math. Soc. (2) 4, 408-410 (1971-1972) · Zbl 0229.47047
[169] Dowling, PN; Turett, B., Coordinatewise star-shaped sets in \(c_0\), J. Math. Anal. Appl., 346, 1, 39-40 (2008) · Zbl 1139.46017
[170] Drešević, M., A note on the Kakutani lemma, Mat. Vesnik, 7, 22, 347-348 (1970) · Zbl 0211.54301
[171] Drešević, M., A certain generalization of Blaschke’s theorem to the class of \(m\)-convex sets (Serbo-Croatian, English summary), Mat. Vesnik, 7, 22, 223-226 (1970) · Zbl 0198.27101
[172] Eckhoff, J.; Gruber, PM; Wills, JM, Helly, Radon, and Carathéodory type theorems, Handbook of Convexity, 389-448 (1993), Amsterdam: North-Holland, Amsterdam · Zbl 0791.52009
[173] Edelsbrunner, H.; Preparata, FP, Minimum polygonal separation, Inform. Comput., 77, 3, 218-232 (1988) · Zbl 0642.52004
[174] Edelstein, M.: On some aspects of fixed point theory in Banach spaces. In: The Geometry of Metric and Linear Spaces (Proceeding Conference, Michigan State University, East Lansing, Michigan, 1974), pp. 84-90. Lecture Notes in Mathematics., vol. 490, Springer, Berlin (1975) · Zbl 0316.47036
[175] Edelstein, M.; Keener, L., Characterizations of infinite-dimensional and nonreflexive spaces, Pac. J. Math., 57, 365-369 (1975) · Zbl 0289.46011
[176] Edelstein, M.; Keener, L.; O’Brien, R., On points at which a set is cone-shaped, Proc. Am. Math. Soc., 66, 2, 327-330 (1977) · Zbl 0367.46017
[177] ElGindy, H.; Toussaint, GT, On geodesic properties of polygons relevant to linear time triangulation, Vis. Comput., 5, 68-74 (1989) · Zbl 0668.65022
[178] El-Sayied, HK, On \(D\)-starshaped sets, Far East J. Math. Sci. (FJMS), 28, 2, 469-481 (2008) · Zbl 1149.53011
[179] Erdös, P., Gruber, P.M., Hammer, J.: Lattice Points. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 39. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York (1989) · Zbl 0683.10025
[180] Falconer, K., The dimension of the convex kernel of a compact starshaped set, Bull. Lond. Math. Soc., 9, 313-316 (1977) · Zbl 0374.52006
[181] Falconer, K., On the equireciprocal point problem, Geom. Dedic., 14, 2, 113-126 (1983) · Zbl 0523.52004
[182] Fang, J., A reverse isoperimetric inequality for embedded starshaped plane curves, Arch. Math. (Basel), 108, 6, 621-624 (2017) · Zbl 1365.52009
[183] Fang, Y-P; Huang, N-J, Increasing-along-rays property, vector optimization and well-posedness, Math. Methods Oper. Res., 65, 99-114 (2007) · Zbl 1137.49023
[184] Fardoun, A.; Regbaoui, R., Flow of starshaped Euclidean hypersurfaces by Weingarten curvatures, Palest. J. Math., 6, Special Issue I, 11-36 (2017) · Zbl 1366.53001
[185] Fenchel, W.: Convexity through the ages. In: Convexity and Its Applications, pp. 120-130, Birkhäuser, Basel (1983) · Zbl 0518.52001
[186] Fischer, P.; Slodkowski, Z., Mean value inequalities for convex and star-shaped sets, Aequat. Math., 70, 3, 213-224 (2005) · Zbl 1089.52006
[187] Flåm, SD, A characterizaton of \(\mathbb{R}^2\) by the concept of mild convexity, Pac. J. Math., 79, 2, 371-373 (1978) · Zbl 0396.46007
[188] Florentin, DI; Segal, A., Minkowski symmetrizations of star shaped sets, Geom. Dedic., 184, 115-119 (2016) · Zbl 1372.52011
[189] Foland, N.; Marr, J., Sets with zero dimensional kernels, Pac. J. Math., 19, 429-432 (1966) · Zbl 0173.15304
[190] Forte Cunto, A., Continuity of the visibility function, Publ. Mat., 35, 323-332 (1991) · Zbl 0746.52010
[191] Forte Cunto, A.; Piacquadio Losada, M.; Toranzos, F., The visibility function revisited, J. Geom., 65, 101-110 (1999) · Zbl 0963.52003
[192] Forte Cunto, A., Toranzos, F.: Visibility inside a smooth Jordan domain. Math. Notae 37, 31-41 (1993-1994) · Zbl 0960.52500
[193] Forte Cunto, A.; Toranzos, F.; Piacquadio Losada, M., Low levels of visibility, Bull. Soc. R. Sc. Liège, 70, 23-27 (2001) · Zbl 1005.52005
[194] Formica, A.; Rodríguez, M., Properties and relations between visibility and illumination operators, Notas Mat., 259, 96-104 (2007)
[195] Francini, E.: Starshapedness of level sets for solutions of nonlinear parabolic equations. Rend. Istit. Mat. Univ. Trieste 28(1-2), 49-62 (1996-1997) · Zbl 0928.35024
[196] Francini, E.: Starshapedness of level sets for solutions of elliptic and parabolic equations (Italian). In: Proceedings of the Conference “Differential Equations” (Italian), Ferrara, 1996. Ann. Univ. Ferrara Sez. VII (N.S.) 41 (1996), suppl., pp. 183-188 (1997) · Zbl 0903.35003
[197] Francini, E., Starshapedness of level sets for solutions of nonlinear elliptic equations, Math. Nachr., 193, 49-56 (1998) · Zbl 0908.35007
[198] Francini, E.; Greco, A., Blow-up in exterior domains: existence and star-shapedness, Z. Anal. Anwend., 17, 2, 431-441 (1998) · Zbl 0902.35041
[199] Ganguly, A., An application of a fixed point theorem to approximation theory, J. Indian Acad. Math., 8, 2, 69-70 (1986) · Zbl 0644.41021
[200] Ganguly, A.; Jadhav, HK, An application of fixed point theorem to approximation theory, Pure Appl. Math. Sci., 42, 1-2, 19-22 (1995) · Zbl 0893.47035
[201] Gardner, RJ, \(X\)-rays of polygons, Discrete Comput. Geom., 7, 3, 281-293 (1992) · Zbl 0748.52003
[202] Gardner, RJ, Intersection bodies and the Busemann-Petty problem, Trans. Am. Math. Soc., 342, 1, 435-445 (1994) · Zbl 0801.52005
[203] Gardner, RJ, On the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies, Bull. Am. Math. Soc. (N.S.), 30, 2, 222-226 (1994) · Zbl 0814.52003
[204] Gardner, R.J.: Geometric Tomography. Second edition. Encyclopedia of Mathematics and its Applications, vol. 58. Cambridge University Press, New York (2006) · Zbl 1102.52002
[205] Gardner, RJ, The dual Brunn-Minkowski theory for bounded Borel sets: dual affine quermassintegrals and inequalities, Adv. Math., 216, 1, 358-386 (2007) · Zbl 1126.52008
[206] Gardner, RJ; Hug, D.; Weil, W., Operations between sets in geometry, J. Eur. Math. Soc., 15, 6, 2297-2352 (2013) · Zbl 1282.52006
[207] Gardner, RJ; Hug, D.; Weil, W.; Xing, S.; Ye, D., General volumes in the Orlicz-Brunn-Minkowski theory and a related Minkowski problem I, Calc. Var. Partial Differ. Equ., 58, 1, Art. 12, 35 (2019) · Zbl 1404.52004
[208] Gardner, RJ; Hug, D.; Weil, W.; Ye, D., The dual Orlicz-Brunn-Minkowski theory, J. Math. Anal. Appl., 430, 2, 810-829 (2015) · Zbl 1320.52008
[209] Gardner, RJ; Koldobsky, A.; Schlumprecht, T., An analytic solution to the Busemann-Petty problem on sections of convex bodies, Ann. Math. (2), 149, 2, 691-703 (1999) · Zbl 0937.52003
[210] Gardner, RJ; Soranzo, A.; Volčič, A., On the determination of star and convex bodies by section functions, Discrete Comput. Geom., 21, 1, 69-85 (1999) · Zbl 0923.52003
[211] Gardner, RJ; Volčič, A., Tomography of convex and star bodies, Adv. Math., 108, 2, 367-399 (1994) · Zbl 0831.52002
[212] Gasinski, L.; Liu, Z.; Migórski, St; Ochal, A.; Peng, Z., Hemivariational inequality approach to evolutionary constrained problems on star-shaped sets, J. Optim. Theory Appl., 164, 2, 514-533 (2015) · Zbl 1440.47048
[213] Gdawiec, K., Star-shaped set inversion fractals, Fractals, 22, 4, 1450009, 7 pp (2014)
[214] Gergen, JJ, Note on the Green function of a star-shaped three dimensional region, Am. J. Math., 53, 746-752 (1931) · Zbl 0003.00802
[215] Gerhardt, C., Flow of nonconvex hypersurfaces into spheres, J. Differ. Geom., 32, 1, 299-314 (1990) · Zbl 0708.53045
[216] Gerhardt, C., Inverse curvature flows in hyperbolic space, J. Differ. Geom., 89, 3, 487-527 (2011) · Zbl 1252.53078
[217] Ghomi, M., Torsion of locally convex curves, Proc. Am. Math. Soc., 147, 4, 1699-1707 (2019) · Zbl 1409.53007
[218] Girardi, M., Multiple orbits for Hamiltonian systems on starshaped surfaces with symmetries, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1, 4, 285-294 (1984) · Zbl 0582.70019
[219] Göhde, D., Elementare Bemerkungen zu nichtexpansiven Selbstabbildungen nicht konvexer Mengen im Hilbertraum, Math. Nachr., 63, 331-335 (1974) · Zbl 0307.47053
[220] González, V.; Rodríguez, M., Some geometrical results about the convex deficiency of a compact set, Appl. Math. Sci., 2, 15, 719-723 (2008) · Zbl 1175.52005
[221] Goodey, P., A note on starshaped sets, Pac. J. Math., 61, 1, 151-152 (1975) · Zbl 0298.52008
[222] Goodey, P.; Lutwak, E.; Weil, W., Functional analytic characterizations of classes of convex bodies, Math. Z., 222, 3, 363-381 (1996) · Zbl 0874.52001
[223] Goodey, P.; Weil, W., Intersection bodies and ellipsoids, Mathematika, 42, 2, 295-304 (1995) · Zbl 0835.52009
[224] Goodey, P.; Weil, W., Average functions for star-shaped sets, Adv. Appl. Math., 36, 70-84 (2006) · Zbl 1094.52003
[225] Gorokhovik, V.V.: On the star-shapedness of sets at infinity (Russian, English and Russian summary). Vest Nats Akad Navuk Belarusi Ser. Fiz.-Mat. Navuk (2) 5-8, 139 (2001)
[226] Góźdź, S., Star-shaped curves with constant largenesses \(L_{h(z)}\), Facta Univ. Ser. Math. Inf., 4, 75-82 (1989) · Zbl 0697.53007
[227] Grinberg, EL; Quinto, ET, Analytic continuation of convex bodies and Funk’s characterization of the sphere, Pac. J. Math., 201, 2, 309-322 (2001) · Zbl 1090.52004
[228] Grinberg, EL; Zhang, GY, Convolutions, transform, and convex bodies, Proc. Lond. Math. Soc. (3), 78, 1, 77-115 (1999) · Zbl 0974.52001
[229] Groemer, H., Stability results for convex bodies and related spherical integral transformations, Adv. Math., 109, 1, 45-74 (1994) · Zbl 0821.52002
[230] Groemer, H.: Geometric Applications of Fourier Series and Spherical Harmonics. Encyclopedia of Mathematics and its Applications, vol. 61. Cambridge University Press, Cambridge (1996) · Zbl 0877.52002
[231] Groemer, H., On a spherical integral transformation and sections of star bodies, Monatsh. Math., 126, 2, 117-124 (1998) · Zbl 0918.52002
[232] Gruber, P.M.: Zur Geschichte der Konvexgeometrie und der Geometrie der Zahlen. Ein Jahrhundert Mathematik: 1890-1990, pp. 421-455, Dokumente Gesch. Math. vol. 6, Friedr. Vieweg, Braunschweig (1990) · Zbl 0864.11004
[233] Gruber, PM; Gruber, PM; Wills, JM, Baire categories in convexity, Handbook of Convex Geometry, 1327-1346 (1993), Amsterdam: North-Holland, Amsterdam · Zbl 0791.52002
[234] Gruber, PM; Gruber, PM; Wills, JM, History of convexity, Handbook of Convex Geometry, 1-15 (1993), Amsterdam: North-Holland, Amsterdam · Zbl 0791.52001
[235] Gruber, P.M.: Convex and Discrete Geometry. Grundlehren der Mathematischen Wissenschaften, vol. 336. Springer, Berlin (2007) · Zbl 1139.52001
[236] Gruber, P.M., Lekkerkerker, C.G.: Geometry of Numbers. Second edition, North-Holland Mathematical Library, vol. 37. North-Holland Publishing Co., Amsterdam (1987) · Zbl 0611.10017
[237] Gruber, PM; Zamfirescu, T., Generic properties of compact starshaped sets, Proc. Am. Math. Soc., 108, 207-214 (1990) · Zbl 0683.52008
[238] Grünbaum, B., Convex Polytopes (1967), Berlin: Wiley, Berlin · Zbl 0163.16603
[239] Grünbaum, B.; Shephard, GC, Isohedra with nonconvex faces, J. Geom., 63, 1-2, 76-96 (1998) · Zbl 0938.51016
[240] Guan, P.; Li, J., The quermassintegral inequalities for \(k\)-convex starshaped domains, Adv. Math., 221, 5, 1725-1732 (2009) · Zbl 1170.53058
[241] Guan, P.; Li, J.; Li, Y., Hypersurfaces of prescribed curvature measure, Duke Math. J., 161, 10, 1927-1942 (2012) · Zbl 1254.53073
[242] Guan, P., Shen, X.: A rigidity theorem for hypersurfaces in higher dimensional space forms. In: Analysis, Complex Geometry, and Mathematical Physics (in Honor of Duong H. Phong), pp. 61-65, Contemp. Math., vol. 644. Amer. Math. Soc., Providence (2015) · Zbl 1335.53070
[243] Guay, M.D., Singh, K.L., Whitfield, J.H.M.: Fixed point theorems for nonexpansive mappings in convex metric spaces. In: Nonlinear Analysis and Applications (St. Johns, Nfld., 1981), pp. 179-189, Lecture Notes in Pure and Appl. Math., vol. 80. Dekker, New York (1982) · Zbl 0501.54030
[244] Gueron, S.; Shafrir, I., A weighted Erdős-Mordell inequality for polygons, Am. Math. Monthly, 112, 3, 257-263 (2005) · Zbl 1085.51023
[245] Guerrero-Zarazua, Z.; Jerónimo-Castro, J., Some comments on floating and centroid bodies in the plane, Aequat. Math., 92, 2, 211-222 (2018) · Zbl 1387.52002
[246] Guo, F.; Liu, C., Multiplicity of Lagrangian orbits on symmetric star-shaped hypersurfaces, Nonlinear Anal., 69, 4, 1425-1436 (2008) · Zbl 1153.37407
[247] Guo, F.; Liu, C., Multiplicity of characteristics with Lagrangian boundary values on symmetric star-shaped hypersurfaces, J. Math. Anal. Appl., 353, 1, 88-98 (2009) · Zbl 1180.53079
[248] Haberl, C., \(L_p\) intersection bodies, Adv. Math., 217, 6, 2599-2624 (2008) · Zbl 1140.52003
[249] Haberl, C., Star body valued valuations, Indiana Univ. Math. J., 58, 5, 2253-2276 (2009) · Zbl 1183.52003
[250] Haberl, C., Ludwig, M.: A characterization of \(L_p\) intersection bodies. Int. Math. Res. Not. (2006), Art. ID 10548, 29 p · Zbl 1115.52006
[251] Habiniak, L., Fixed point theorems and invariant approximations, J. Approx. Theory, 56, 3, 241-244 (1989) · Zbl 0673.41037
[252] Halpern, B., The kernel of a starshaped subset of the plane, Proc. Am. Math. Soc., 23, 692-696 (1969) · Zbl 0185.25802
[253] Halpern, B., On the immersion of an \(n\)-dimensional manifold in \((n+1)\)-dimensional Euclidean space, Proc. Am. Math. Soc., 30, 181-184 (1971) · Zbl 0222.57016
[254] Han, F.; Ma, X-N; Wu, D., The existence of \(k\)-convex hypersurface with prescribed mean curvature, Calc. Var. Partial Differ. Equ., 42, 1-2, 43-72 (2011) · Zbl 1233.35074
[255] Hansen, G.; Martini, H., On closed starshaped sets, J. Convex Anal., 17, 659-671 (2010) · Zbl 1207.52006
[256] Hansen, G.; Martini, H., Starshapedness vs. convexity, Results Math., 59, 185-197 (2011) · Zbl 1218.52004
[257] Hansen, G.; Martini, H., Dispensable points, radial functions and boundaries of starshaped sets, Acta Sci. Math. (Szeged), 80, 689-699 (2014) · Zbl 1340.52002
[258] Hare, W.; Kenelly, J., Concerning sets with one point kernel, Nieuw Arch. Wisk., 14, 103-105 (1966) · Zbl 0145.19202
[259] Hare, W.; Kenelly, J., Intersections of maximal star-shaped sets, Proc. Am. Math. Soc., 19, 1299-1302 (1968) · Zbl 0174.25301
[260] Haydon, R.; Odell, E.; Sternfeld, Y., A fixed point theorem for a class of star-shaped sets in \(C_0\), Isr. J. Math., 38, 1-2, 75-81 (1981) · Zbl 0473.47045
[261] Helly, E., Über Mengen konvexer Körper mit gemeinschaftlichen Punkten, Jber. Deutsch. Math. Verein., 32, 175-176 (1923) · JFM 49.0534.02
[262] Herburt, I., On convex hulls of star sets, Bull. Polish Acad. Sci. Math., 49, 4, 433-440 (2001) · Zbl 0996.52004
[263] Herburt, I, Moszyńska, M., Pronk, D.: Fractal star bodies. In: Convex and Fractal Geometry, pp. 149-171. Banach Center Publications, vol. 84. Polish Acad. Sci. Inst. Math., Warsaw (2009) · Zbl 1184.28005
[264] Hiripitiyage, KL; Yaskin, V., On sections of convex bodies in hyperbolic space, J. Math. Anal. Appl., 445, 2, 1394-1409 (2017) · Zbl 1348.52007
[265] Hirose, T., On the convergence theorem for star-shaped sets in \(\mathbb{R}^n \), Proc. Jpn. Acad., 41, 3, 209-211 (1965) · Zbl 0135.22605
[266] Ho, ChW, Deforming star-shaped polygons in the plane, Geom. Dedic., 9, 4, 451-460 (1980) · Zbl 0446.52011
[267] Ho, ChW, Deforming star-shaped polygons in the plane. II, Bull. Inst. Math. Acad. Sin., 9, 3, 347-357 (1981) · Zbl 0485.52004
[268] Horn, A.; Valentine, FA, Some properties of \(L\)-sets in the plane, Duke Math. J., 16, 131-140 (1949) · Zbl 0034.10403
[269] Horst, R.; Pardalos, P.; Thoai, N., Introduction to Global Optimization (1995), Berlin: Kluwer, Berlin · Zbl 0836.90134
[270] Horvath, CD; Lassonde, M., Intersection of sets with \(n\)-connected unions, Proc. Am. Math. Soc., 125, 1209-1214 (1997) · Zbl 0861.54020
[271] Howard, R.; Treibergs, A., A reverse isoperimetric inequality, stability and extremal theorems for plane curves with bounded curvature, Rocky Mountain J. Math., 25, 2, 635-684 (1995) · Zbl 0909.53002
[272] Hu, R.: Lower convergence of minimal sets in star-shaped vector optimization problems. J. Appl. Math. (2014), Art. ID 532195, 7 p · Zbl 1442.49029
[273] Hu, T.; Heng, W-S, An extension of Markov-Kakutani’s fixed point theorem, Indian J. Pure Appl. Math., 32, 6, 899-902 (2001) · Zbl 1192.47049
[274] Hu, X.; Long, Y., Closed characteristics on non-degenerate star-shaped hypersurfaces in \(\mathbb{R}^{2n} \), Sci. China Ser. A, 45, 8, 1038-1052 (2002) · Zbl 1099.37047
[275] Huisken, G.; Ilmanen, T., Higher regularity of the inverse mean curvature flow, J. Differ. Geom., 80, 3, 433-451 (2008) · Zbl 1161.53058
[276] Hummel, JA, Multivalent starlike functions, J. Anal. Math., 18, 133-160 (1967) · Zbl 0146.30403
[277] Hussain, N.; Khan, AR, Common fixed-point results in best approximation theory, Appl. Math. Lett., 16, 4, 575-580 (2003) · Zbl 1063.47055
[278] Isakov, V.: Inverse Source Problems. Mathematical Surveys and Monographs, vol. 34. American Mathematical Society (AMS), Providence (1990) · Zbl 0721.31002
[279] Ivochkina, NM; Nehring, T.; Tomi, F., Evolution of starshaped hypersurfaces by nonhomogeneous curvature functions, St. Petersburg Math. J., 12, 1, 145-160 (2001) · Zbl 1031.53093
[280] Jahn, T.; Martini, H.; Richter, C., Bi- and multifocal curves and surfaces for gauges, J. Convex Anal., 23, 733-774 (2016) · Zbl 1348.51006
[281] Jin, H.; Yuan, S.; Leng, G., On the dual Orlicz mixed volumes, Chin. Ann. Math. Ser. B, 36, 6, 1019-1026 (2015) · Zbl 1331.52005
[282] Jin, Q.; Li, Y., Starshaped compact hypersurfaces with prescribed \(k\)-th mean curvature in hyperbolic space, Discrete Contin. Dyn. Syst., 15, 2, 367-377 (2006) · Zbl 1177.53058
[283] Jongmans, F.: Etude des cônes associés à un ensemble. Séminaire stencilé, Liège (1983-1984)
[284] Kalashnikov, VV; Talman, AJJ; Alanis-Lopez, L.; Kalashnykova, N., Extended antipodal theorems, J. Optim. Theory Appl., 177, 2, 399-412 (2018) · Zbl 1433.47002
[285] Kawohl, B., Starshapedness of level sets for the obstacle problem and for the capacitory potential problem, Proc. Am. Math. Soc., 89, 4, 637-640 (1983) · Zbl 0556.35051
[286] Kawohl, B.: Rearrangements and convexity of level sets in PDE. Lecture Notes in Mathematics, vol. 1150. Springer, Berlin (1985) · Zbl 0593.35002
[287] Kawohl, B., On starshaped rearrangement and applications, Trans. Am. Math. Soc., 296, 1, 377-386 (1986) · Zbl 0612.35033
[288] Kawohl, B.: Geometrical properties of level sets of solutions to elliptic problems. In: Nonlinear Functional Analysis and its Applications, Part 2 (Berkeley, California, 1983), pp. 25-36, Proc. Sympos. Pure Math., vol. 45, Part 2. Amer. Math. Soc., Providence (1986) · Zbl 0597.35016
[289] Kenelly, J.; Hare, W.; Evans, B.; Ludescher, W., Convex components, extreme points and the convex kernel, Proc. Am. Math. Soc., 21, 83-87 (1969) · Zbl 0174.25303
[290] Keogh, FR, Some inequalities for convex and star-shaped domains, J. Lond. Math. Soc., 29, 121-123 (1954) · Zbl 0056.30003
[291] Kilicman, A., Saleh, W.: A note on starshaped sets in 2-dimensional manifolds without conjugate points. J. Funct. Spaces (2014), Art. ID 675735, 3 p · Zbl 1296.53068
[292] Kjeldsen, TH, From measuring tool to geometrical object: Minkowski’s development of the concept of convex bodies, Arch. Hist. Exact Sci., 62, 59-89 (2008) · Zbl 1145.01014
[293] Klain, DA, Star valuations and dual mixed volumes, Adv. Math., 121, 1, 80-101 (1996) · Zbl 0858.52003
[294] Klain, DA, Invariant valuations on star-shaped sets, Adv. Math., 125, 95-113 (1997) · Zbl 0889.52007
[295] Klain, DA, An error estimate for the isoperimetric deficit, Ill. J. Math., 49, 3, 981-992 (2005) · Zbl 1089.52007
[296] Klee, VL, Extremal structure of convex sets, Arch. Math., 8, 234-240 (1957) · Zbl 0079.12501
[297] Klee, VL, Extremal structure of convex sets II, Math. Z., 69, 90-104 (1958) · Zbl 0079.12502
[298] Klee, VL, Convex sets in linear spaces, Duke Math. J., 18, 443-466 (1951) · Zbl 0042.36201
[299] Klee, VL, The critical set of a convex body, Am. J. Math., 75, 178-188 (1953) · Zbl 0050.16604
[300] Klee, V.L.: Relative extreme points. Proc. 1961 Internat. Sympos. Linear Spaces (Jerusalem 1960), pp. 282-289. Jerusalem Academic Press; Pergamon, Oxford; Jerusalem (1961) · Zbl 0173.41103
[301] Klee, VL, A theorem on convex kernels, Mathematika, 12, 89-93 (1965) · Zbl 0137.41601
[302] Klein, R.: Concrete and Abstract Voronoĭ Diagrams. Lecture Notes in Computer Science, vol. 400. Springer, Berlin (1989) · Zbl 0699.68005
[303] Koch, CF; Marr, JM, A characterization of unions of two star-shaped sets, Proc. Am. Math. Soc., 17, 1341-1343 (1966) · Zbl 0152.20703
[304] Koldobsky, A., Intersection bodies, positive definite distributions, and the Busemann-Petty problem, Am. J. Math., 120, 4, 827-840 (1998) · Zbl 0914.52001
[305] Koldobsky, A., The Busemann-Petty problem via spherical harmonics, Adv. Math., 177, 1, 105-114 (2003) · Zbl 1034.52005
[306] Koldobsky, A.: Sections of star bodies and the Fourier transform. In: Harmonic Analysis at Mount Holyoke (South Hadley, MA, 2001), pp. 225-247, Contemp. Math., vol. 320. Amer. Math. Soc., Providence (2003) · Zbl 1055.52003
[307] Koldobsky, A.: Fourier Analysis in Convex Geometry. Mathematical Surveys and Monographs, vol. 116. American Mathematical Society, Providence (2005) · Zbl 1082.52002
[308] Koldobsky, A., Stability and separation in volume comparison problems, Math. Model. Nat. Phenom., 8, 1, 156-169 (2013) · Zbl 1267.52006
[309] Koldobsky, A.; Paouris, G.; Zymonopoulou, M., Complex intersection bodies, J. Lond. Math. Soc. (2), 88, 2, 538-562 (2013) · Zbl 1279.52007
[310] Koldobsky, A., Yaskin, V.: The Interface Between Convex Geometry and Harmonic Analysis. CBMS Regional Conference Series in Mathematics, vol. 108. Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence (2008) · Zbl 1139.52010
[311] Kołodziejczyk, K., A refinement of Valentine’s theorem, Arch. Math. (Basel), 43, 3, 270-274 (1984) · Zbl 0595.52010
[312] Kołodziejczyk, K., Starsapedness in convexity spaces, Compos. Math., 56, 3, 361-367 (1985) · Zbl 0584.52001
[313] Kołodziejczyk, K., On starshapedness of the union of closed sets in \(\mathbb{R}^n \), Colloq. Math., 53, 2, 193-197 (1987) · Zbl 0635.52006
[314] Kołodziejczyk, K., The starshapedness number and a Krasnosel’skiĭ-type theorem in a convexity space, Arch. Math. (Basel), 49, 6, 535-544 (1987) · Zbl 0636.52002
[315] Kołodziejczyk, K., Krasnosel’ski-type parameters of convexity spaces, Rev. Un. Mat. Argentina, 40, 3-4, 93-102 (1997) · Zbl 0912.52002
[316] Kosiński, A., Note on star-shaped sets, Proc. Am. Math. Soc., 13, 931-933 (1962) · Zbl 0111.35305
[317] Kovalev, MD, The smallest Lebesgue covering exists, Math. Notes, 40, 736-739 (1986) · Zbl 0638.52004
[318] Krasnosel’skii, MA, Sur un critère pour qu’un domaine soit étoilé, Mat. Sb., 19, 309-310 (1946) · Zbl 0061.37705
[319] Krein, M.; Milman, D., Extreme points of regularly convex sets, Stud. Math., 9, 133-138 (1940) · JFM 66.0533.01
[320] Kroó, A., Density of multivariate homogeneous polynomials on star like domains, J. Math. Anal. Appl., 469, 1, 239-251 (2019) · Zbl 1402.41001
[321] Kuhfittig, P., Fixed-point theorems for mappings with non-convex domain and range, Rocky Mountain J. Math., 7, 1, 141-145 (1977) · Zbl 0346.47047
[322] Larman, DG, On the convex kernel of a compact set, Proc. Camb. Phil. Soc., 63, 311-313 (1967) · Zbl 0156.36104
[323] Larman, DG, On the union of two starshaped sets, Pac. J. Math., 21, 521-524 (1967) · Zbl 0148.36902
[324] Lee, DT; Preparata, FP, An optimal algorithm for finding the kernel of a polygon, J. Assoc. Comput. Mach., 26, 3, 415-421 (1979) · Zbl 0403.68051
[325] Leichtweiss, K., Konvexe Mengen (1980), Berlin: Deutscher Verlag der Wissenschaften, Berlin · Zbl 0427.52001
[326] Li, H.; Wei, Y.; Xiong, C., A note on Weingarten hypersurfaces in the warped product manifold, Int. J. Math., 25, 14, 1450121, , 13 pp (2014) · Zbl 1325.53078
[327] Li, Y.; Oliker, V., Starshaped compact hypersurfaces with prescribed \(m\)-th mean curvature in elliptic space, J. Partial Differ. Equ., 15, 3, 68-80 (2002) · Zbl 1183.53054
[328] Li, Y., Wang, W.: The \(L_p\)-dual mixed geominimal surface area for multiple star bodies. J. Inequal. Appl. (2014), Paper No. 456, 10 p · Zbl 1335.52009
[329] Li, Y., Wang, W.: General \(L_p\)-mixed chord integrals of star bodies. J. Inequal. Appl. 2016, Paper No. 58, 12 p · Zbl 1333.52003
[330] Lin, L.; Xiao, L., Modified mean curvature flow of star-shaped hypersurfaces in hyperbolic space, Commun. Anal. Geom., 20, 5, 1061-1096 (2012) · Zbl 1270.53086
[331] Liu, C.; Long, Y., Hyperbolic characteristics on star-shaped hypersurfaces, Ann. Inst. Henri Poincaré, Anal. Non Liné aire, 16, 6, 725-746 (1999) · Zbl 0988.37078
[332] Longinetti, M., A maximum principle for the starshape of solutions of nonlinear Poisson equations, Boll. Un. Mat. Ital. A (6), 4, 1, 91-96 (1985) · Zbl 0566.35050
[333] Lu, F.; Mao, W., On dual Knesser-Süss inequalities, Int. J. Modern Math., 5, 1, 109-117 (2010) · Zbl 1195.52001
[334] Ludwig, M., Intersection bodies and valuations, Am. J. Math., 128, 1409-1428 (2006) · Zbl 1115.52007
[335] Ludwig, M., Valuations on function spaces, Adv. Geom., 11, 4, 745-756 (2011) · Zbl 1235.52023
[336] Lutwak, E., Dual mixed volumes, Pac. J. Math., 58, 2, 531-538 (1975) · Zbl 0273.52007
[337] Lutwak, E., Intersection bodies and dual mixed volumes, Adv. Math., 71, 2, 232-261 (1988) · Zbl 0657.52002
[338] Lutwak, E., Centroid bodies and dual mixed volumes, Proc. Lond. Math. Soc. (3), 60, 2, 365-391 (1990) · Zbl 0703.52005
[339] Lutwak, E.; Yang, D.; Zhang, G., \(L_p\) affine isoperimetric inequalities, J. Differ. Geom., 56, 1, 111-132 (2000) · Zbl 1034.52009
[340] Lutwak, E.; Yang, D.; Zhang, G., The Cramer-Rao inequality for star bodies, Duke Math. J., 112, 59-81 (2002) · Zbl 1021.52008
[341] Lutwak, E.; Yang, D.; Zhang, G., Orlicz projection bodies, Adv. Math., 223, 1, 220-242 (2010) · Zbl 1437.52006
[342] Lutwak, E.; Yang, D.; Zhang, G., Orlicz centroid bodies, J. Differ. Geom., 84, 2, 365-387 (2010) · Zbl 1206.49050
[343] Lv, S.; Leng, G., Cross \(i\)-sections of star bodies and dual mixed volumes, Proc. Am. Math. Soc., 135, 10, 3367-3373 (2007) · Zbl 1130.52004
[344] Magazanik, E.; Perles, MA, Staircase connected sets, Discrete Comput. Geom., 37, 4, 587-599 (2007) · Zbl 1139.52004
[345] Magazanik, E.; Perles, MA, Intersections of maximal staircase sets, J. Geom., 88, 1-2, 127-133 (2008) · Zbl 1140.52002
[346] Mahler, K., Note on lattice points in star domains, J. Lond. Math. Soc., 17, 130-133 (1942) · Zbl 0028.11203
[347] Mahler, K., On lattice points in an infinite star domain, J. Lond. Math. Soc., 18, 233-238 (1943) · Zbl 0060.11802
[348] Mahler, K., Lattice points in two-dimensional star domains. I, Proc. Lond. Math. Soc. (2), 49, 128-157 (1946) · Zbl 0060.11712
[349] Mahler, K.: Lattice points in n-dimensional star bodies. II. Reducibility theorems. I, II, Nederl. Akad. Wetensch. Proc. 49 (1946), 331-343, 444-454 (Indagationes Math. 8 (1946), 200-212, 299-309.) · Zbl 0060.11711
[350] Mahler, K.: Lattice points in n-dimensional star bodies. II. Reducibility theorems. III, IV, Nederl. Akad. Wetensch. Proc. 49, 524-532, 622-631 (1946) (Indagationes Math. 8 (1946), 343-351, 381-390.) · Zbl 0060.11711
[351] Mahler, K., Lattice points in two-dimensional star domains. II, Proc. Lond. Math. Soc. (2), 49, 158-167 (1946) · Zbl 0060.11712
[352] Mahler, K., Lattice points in two-dimensional star domains. II, Proc. Lond. Math. Soc. (2), 49, 168-183 (1946) · Zbl 0060.11712
[353] Mahler, K., On lattice points in \(n\)-dimensional star bodies. I. Existence theorems, Proc. R. Soc. Lond. Ser. A, 187, 151-187 (1946) · Zbl 0060.11710
[354] Mahler, K., Lattice points in \(n\)-dimensional star bodies, Univ. Nac. Tucumán. Revista A, 5, 113-124 (1946) · Zbl 0060.11801
[355] Mahler, K., Über die konvexen Köorper, die sich einem Sternkörper einbeschreiben lassen, Math. Z., 66, 25-33 (1956) · Zbl 0071.27402
[356] Makai, E. Jr; Martini, H., On bodies associated with a given convex body, Can. Math. Bull., 39, 4, 448-459 (1996) · Zbl 0884.52005
[357] Makai, E. Jr; Martini, H., The cross-section body, plane sections of convex bodies and approximation of convex bodies. I, Geom. Dedic., 63, 3, 267-296 (1996) · Zbl 0865.52005
[358] Makai, E. Jr; Martini, H., The cross-section body, plane sections of convex bodies and approximation of convex bodies. II, Geom. Dedic, 70, 3, 283-303 (1998) · Zbl 0914.52002
[359] Makai, E. Jr; Martini, H.; Ödor, T., Maximal sections and centrally symmetric bodies, Mathematika, 47, 19-30 (2000) · Zbl 1012.52008
[360] Makai, E. Jr; Martini, H.; Ódor, T., On a theorem of D. Ryabogin and V. Yaskin about detecting symmetry, Note Mat., 34, 2, 1-5 (2014) · Zbl 1381.52011
[361] Makazhanova, T.K.: On the structure of some classes of star-shaped sets. In: The Theory of Algebraic Structures, Collect. Sci. Works, pp. 60-65, Karaganda Gos. Univ., Karaganda (1985) (in Russian) · Zbl 0643.52003
[362] Makeev, VV, On quadrangles inscribed in a closed curve (Russian), Mat. Zametki, 57, 1, 129-132 (1995) · Zbl 0862.57007
[363] Marcus, M., Transformations of domains in the plane and applications in the theory of functions, Pac. J. Math., 14, 613-626 (1964) · Zbl 0143.29803
[364] Margulis, AS, Equivalence and uniqueness in an inverse problem of potential for homogeneous star-shaped bodies, Dokl. Akad. Nauk SSSR, 312, 3, 577-580 (1990)
[365] Marquardt, T., Inverse mean curvature flow for star-shaped hypersurfaces evolving in a cone, J. Geom. Anal., 23, 3, 1303-1313 (2013) · Zbl 1317.53087
[366] Martini, H.: Cross-sectional measures. Intuitive geometry (Szeged, 1991), pp. 269-310. Colloq. Math. Soc. János Bolyai, vol. 63. North-Holland, Amsterdam (1994) · Zbl 0820.52003
[367] Martini, H.; Richter, C.; Spirova, M., Illuminating and covering convex bodies, Discrete Math., 337, 106-118 (2014) · Zbl 1303.52009
[368] Martini, H.; Soltan, V., Combinatorial problems on the illumination of convex bodies, Aequat. Math., 57, 2-3, 121-152 (1999) · Zbl 0937.52006
[369] Martini, H.; Spirova, M.; Strambach, K., Geometric algebra of strictly convex Minkowski planes, Aequat. Math., 88, 1-2, 49-66 (2014) · Zbl 1320.46015
[370] Martini, H.; Wenzel, W., A characterization of convex sets via visibility, Aequat. Math., 64, 128-135 (2002) · Zbl 1013.52005
[371] Martini, H.; Wenzel, W., An analogue of the Krein-Milman theorem for star-shaped sets, Beitr. Algebra Geom., 44, 441-449 (2003) · Zbl 1043.52006
[372] Martini, H.; Wenzel, W., Illumination and visibility problems in terms of closure operators, Beitr. Algebra Geom., 45, 607-614 (2004) · Zbl 1074.52001
[373] Martino, V.; Montanari, A., Integral formulas for a class of curvature PDE’s and applications to isoperimetric inequalities and to symmetry problems, Forum Math., 22, 2, 255-267 (2010) · Zbl 1198.32016
[374] Massa, S.; Roux, D.; Singh, SP, Fixed point theorems for multifunctions, Indian J. Pure Appl. Math., 18, 6, 512-514 (1987) · Zbl 0631.47040
[375] Mazurenko, SS, A differential equation for the gauge function of the star-shaped attainability set of a differential inclusion, Dokl. Akad. Nauk, 445, 2, 139-142 (2012) · Zbl 1357.93008
[376] McMullen, P., Sets homothetic to intersections of their translates, Mathematika, 25, 2, 264-269 (1978) · Zbl 0399.52005
[377] McMullen, P.: Nondiscrete regular honeycombs. In: Quasicrystals, Networks, and Molecules of Fivefold Symmetry, Ed. I. Hargittai, VCH Verlagsgesellschaft mbH, pp. 159-179, Weinheim (1990)
[378] Melzak, ZA, A class of star-shaped bodies, Can. Math. Bull., 2, 175-180 (1959) · Zbl 0090.12801
[379] Menger, K., Untersuchungen über allgemeine Metrik, I, II, III. Math. Ann., 100, 75-163 (1928) · JFM 54.0622.02
[380] Meyer, M., Maximal hyperplane sections of convex bodies, Mathematika, 46, 1, 131-136 (1999) · Zbl 0988.52019
[381] Michael, TS; Pinciu, V., The art gallery theorem, revisited, Am. Math. Monthly, 123, 8, 802-807 (2016) · Zbl 1391.52010
[382] Mohebi, H.; Naraghirad, E., Cone-separation and star-shaped separability with applications, Nonlinear Anal., 69, 8, 2412-2421 (2008) · Zbl 1190.90124
[383] Mohebi, H.; Sadeghi, H.; Rubinov, AM, Best approximation in a class of normed spaces with star-shaped cone, Numer. Funct. Anal. Optim., 27, 3-4, 411-436 (2006) · Zbl 1098.41036
[384] Molchanov, I., Convex and star-shaped sets associated with multivariate stable distributions, J. Multivar. Anal., 100, 10, 2195-2213 (2009) · Zbl 1196.60029
[385] Molnár, J., Über eine Vermutung von G, Hajos. Acta Math. Hungar., 8, 311-314 (1957) · Zbl 0079.16301
[386] Molnár, J., Über Sternpolygone, Publ. Math. Debrecen, 5, 241-245 (1958) · Zbl 0083.38403
[387] Mordell, LJ, On numbers represented by binary cubic forms, Proc. Lond. Math. Soc. (2), 48, 198-228 (1943) · Zbl 0060.12002
[388] Moszyńska, M., Selected Topics in Convex Geometry (2006), Berlin: Birkhäuser, Berlin
[389] Moszyńska, M., Quotient star bodies, intersection bodies, and star duality, J. Math. Anal. Appl., 232, 45-60 (1999) · Zbl 0928.54007
[390] Moszyńska, M., Looking for selectors of star bodies, Geom. Dedic., 83, 131-147 (2000) · Zbl 0980.52003
[391] Moszyńska, M., On directly additive selectors for convex and star bodies, Glas. Mat. Ser. III, 39(59), 1, 145-154 (2004) · Zbl 1068.52005
[392] Moszyńska, M.; Richter, W-D, Reverse triangle inequality Antinorms and semi-antinorms, Stud. Sci. Math. Hungar., 49, 120-138 (2012) · Zbl 1299.26042
[393] Moszyńska, M.; Sójka, G., Concerning sets of the first Baire category with respect to different metrics, Bull. Pol. Acad. Sci. Math., 58, 1, 47-54 (2010) · Zbl 1207.54039
[394] Mukherjee, RN; Mishra, SK, Multiobjective programming with semilocally convex functions, J. Math. Anal. Appl., 199, 2, 409-424 (1996) · Zbl 0864.90111
[395] Müller, G.; Reinermann, J., Eine Charakterisierung strikt-konvexer Banach-Räume über einen Fixpunktsatz für nichtexpansive Abbildungen, Math. Nachr., 93, 239-247 (1979) · Zbl 0434.47048
[396] Myroshnychenko, S.; Ryabogin, D.; Saroglou, C., Star bodies with completely symmetric sections, Int. Math. Res. Not. IMRN, 10, 3015-3031 (2019) · Zbl 1419.52005
[397] Naraghirad, E., Lin, L.-J.: Strong convergence theorems for generalized nonexpansive mappings on starshaped set with applications. Fixed Point Theory Appl. 2014, Paper No. 72, 24 pp · Zbl 1332.47053
[398] Nashine, HK, An application of a fixed-point theorem to best approximation for generalized affine mapping, Math. Proc. R. Ir. Acad., 107A, 2, 131-136 (2007) · Zbl 1225.47068
[399] Novikov, PS, On the uniqueness for the inverse problem of potential theory, Dokl. Akad. Nauk SSSR (N.S.), 18, 165-168 (1938) · Zbl 0018.30901
[400] Oliker, VI, Hypersurfaces in \(\mathbb{R}^{n+1}\) with prescribed Gaussian curvature and related equations of Monge-Ampère type, Commun. Partial Differ. Equ., 9, 8, 807-838 (1984) · Zbl 0559.58031
[401] Opfer, G., New extremal properties for constructing conformal mappings, Numer. Math., 32, 4, 423-429 (1979) · Zbl 0432.30007
[402] O’Rourke, J., Art Gallery Theorems and Algorithms. International Series of Monographs on Computer Science (1987), New York: The Clarendon Press, New York · Zbl 0653.52001
[403] O’Regan, D.; Shahzad, N., Invariant approximations for generalized I-contractions, Numer. Funct. Anal. Optim., 26, 4-5, 565-575 (2005) · Zbl 1084.41023
[404] Pan, S.; Zhang, H.; Zhang, L., Star-shaped differentiable functions and star-shaped differentials, Commun. Math. Res., 26, 1, 41-52 (2010) · Zbl 1224.49020
[405] Pankrashkin, Ko, An inequality for the maximum curvature through a geometric flow, Arch. Math. (Basel), 105, 3, 297-300 (2015) · Zbl 1325.53008
[406] Park, J-H; Shin, SY; Chwa, K-Y; Woo, TC, On the number of guard edges of a polygon, Discrete Comput. Geom., 10, 4, 447-462 (1993) · Zbl 0788.52005
[407] Park, S., Fixed points on star-shaped sets, Nonlinear Anal. Forum, 6, 2, 275-279 (2001) · Zbl 1011.47047
[408] Park, S.; Yoon, J., Remarks on fixed point theorems on star-shaped sets, J. Korean Math. Soc., 18, 135-140 (1982) · Zbl 0485.54035
[409] Pasquotto, F., A short history of the Weinstein conjecture, Jahresber. Dtsch. Math. Ver., 114, 3, 119-130 (2012) · Zbl 1263.37003
[410] Peck, NT, Support points in locally convex spaces, Duke Math. J., 38, 271-278 (1971) · Zbl 0213.39103
[411] Penot, J-P, A duality for starshaped functions, Bull. Polish Acad. Sci. Math., 50, 2, 127-139 (2002) · Zbl 1021.46007
[412] Penot, J-P, The directional subdifferential of the difference of two convex functions, J. Glob. Optim., 49, 3, 505-519 (2011) · Zbl 1223.26030
[413] Peterson, B.: Is there a Krasnosel’skii theorem for finitely starlike sets? Convexity and Related Combinatorial Geometry, pp. 81-84, Marcel Dekker, New York (1982) · Zbl 0483.52006
[414] Petty, CM, Centroid surfaces, Pac. J. Math., 11, 1535-1547 (1961) · Zbl 0103.15604
[415] Piacquadio Losada, M.; Forte Cunto, A.; Toranzos, FA, Continuity of the visibility function in the boundary, Geom. Dedic., 80, 43-49 (2000) · Zbl 1039.52007
[416] Pinkall, U., Hamiltonian flows on the space of star-shaped curves, Results Math., 27, 3-4, 328-332 (1995) · Zbl 0835.35128
[417] Post, K., Star extension of plane convex sets, Indag. Math., 26, 330-338 (1964) · Zbl 0125.11204
[418] Preparata, F.; Shamos, M., Computational Geometry: An Introduction (1985), Berlin: Springer, Berlin · Zbl 0759.68037
[419] Qiu, D.; Shu, L.; Mo, Z-W, On starshaped fuzzy sets, Fuzzy Sets Syst., 160, 1563-1577 (2009) · Zbl 1184.03053
[420] Rabinowitz, PH, Periodic solutions of Hamiltonian systems, Commun. Pure Appl. Math., 31, 157-184 (1978) · Zbl 0358.70014
[421] Ramos-Guajardo, A., González-Rodríguez, G., Colubi, A., Ferraro, M.B., Blanco-Fernández, Á.: On some concepts related to star-shaped sets. In: The Mathematics of the Uncertain, pp. 699-708, Stud. Syst. Decis. Control, vol. 142, Springer, Cham (2018)
[422] Reich, S.; Zaslavski, A., Nonexpansive set-valued mappings on bounded star-shaped sets, J. Nonlinear Convex Anal., 18, 7, 1383-1392 (2017) · Zbl 1474.47094
[423] Reinermann, J., Fixed point theorems for nonexpansive mappings on starshaped domains, Ber. Ges. Math. Datenverarb. Bonn, 103, 23-28 (1975) · Zbl 0319.47033
[424] Reinermann, J.; Stallbohm, V., Fixed point theorems for compact and nonexpansive mappings on starshaped domains, Commentat. Math. Univ. Carol., 15, 775-779 (1974) · Zbl 0295.47056
[425] Reinermann, J., Stallbohm, V.: Fixed point theorems for compact and nonexpansive mappings on starshaped domains. Papers presented at the 5th Balkan Mathematical Congress (Belgrade, 1974). Math. Balkanica, vol. 4, pp. 511-516 (1974) · Zbl 0314.47030
[426] Ren, L., Xin, J.: Almost global existence for the Neumann problem of quasilinear wave equations outside star-shaped domains in 3D. Electron. J. Differential Equations 2017, Paper No. 312, 22 p · Zbl 1387.35397
[427] Richter, W.-D.: Geometric disintegration and star-shaped distributions. J. Stat. Distrib. Appl., Vol. 1, Art. 20, 2014, 24 p · Zbl 1330.60028
[428] Richter, W.-D., Schicker, K.: Polyhedral star-shaped distributions. J. Probab. Stat. 2017, Art. ID 7176897, 23 p · Zbl 1431.60016
[429] Roberts, AW; Varberg, DE, Convex Functions (1973), Berlin: Academic Press, Berlin · Zbl 0271.26009
[430] Robkin, E.E.: Characterizations of starshaped sets. Ph.D. Thesis, University of California, Los Angeles, 70 p (1965)
[431] Rockafellar, T., Convex Analysis (1970), Princeton: Princeton University Press, Princeton · Zbl 0202.14303
[432] Rodríguez, M.: Extensión de los conceptos de visibilidad afín. Tesis Doctoral, Universidad de Buenos Aires, 1997 (in Spanish, English summary)
[433] Rodríguez, M., Properties of external visibility, Rev. Un. Mat. Argentina, 40, 15-23 (1997) · Zbl 0933.52013
[434] Rodríguez, M., Krasnoselsky-type theorems involving outward rays, Bull. Soc. R. Sc. Liège, 67, 23-30 (1998) · Zbl 0964.52011
[435] Rodríguez, M., Toranzos, F.: Finitely starshaped sets. Proc. Iberoamerican Cong. Geom. (Olmué, Chile), pp. 245-254 (1998)
[436] Rodríguez, M.; Toranzos, F., Structure of closed finitely starshaped sets, Proc. Am. Math. Soc., 128, 1433-1441 (2000) · Zbl 0954.52010
[437] Rodríguez, M.; Toranzos, F., Finite illumination of unbounded closed convex sets, Int. Math. Forum, 1, 27-39 (2006) · Zbl 1149.52004
[438] Rogers, CA, The number of lattice points in a star body, J. Lond. Math. Soc., 26, 307-310 (1951) · Zbl 0043.05102
[439] Rosenfeld, M.; Tan, TN, Weighted Erdős-Mordell inequality for star-shaped polygons, Geombinatorics, 25, 1, 36-44 (2015) · Zbl 1352.51017
[440] Rubin, B., On the determination of star bodies from their half-sections, Mathematika, 63, 2, 462-468 (2017) · Zbl 1375.44002
[441] Rubinov, AM, Abstract Convexity and Global Optimization (2000), Berlin: Kluwer Academic Publishers, Berlin · Zbl 0985.90074
[442] Rubinov, A.M.: Radiant sets and their gauges. In: Quasidifferentiability and Related Topics, pp. 235-261, Nonconvex Optim. Appl., vol. 43. Kluwer Acad. Publ., Dordrecht (2000) · Zbl 0990.90131
[443] Rubinov, AM; Sharikov, EV, Star-shaped separability with applications, J. Convex Anal., 13, 3-4, 849-860 (2006) · Zbl 1142.52007
[444] Rubinov, A.M., Shveidel, A.P.: Separability of star-shaped sets with respect to infinity. In: Progress in Optimization (Perth, 1998), pp. 45-63, Appl. Optim., vol. 39. Kluwer Acad. Publ., Dordrecht (2000) · Zbl 1043.90555
[445] Rubinov, A.M., Yagubov, A.A.: The space of star-shaped sets and its applications in nonsmooth optimization. In: Quasidifferential Calculus. Math. Programming Stud. No. 29, pp. 176-202 (1986) · Zbl 0605.90105
[446] Rubinov, AM; Yagubov, AA, Spaces of sets that are star-shaped in the cone sense (Russian; English and Azerbaijani summary), Akad. Nauk Azerbaĭdzhan. SSR Dokl., 42, 3, 6-9 (1986) · Zbl 0605.46015
[447] Ruppert, J.; Seidel, R., On the difficulty of triangulating three-dimensional nonconvex polyhedra, Discrete Comput. Geom., 7, 3, 227-253 (1992) · Zbl 0747.52009
[448] Ryabogin, D.: On symmetries of projections and sections of convex bodies. In: Discrete Geometry and Symmetry, pp. 297-309, Springer Proc. Math. Stat., vol. 234. Springer, Cham (2018) · Zbl 1421.52006
[449] Ryabogin, D.; Yaskin, V., Detecting symmetry in star bodies, J. Math. Anal. Appl., 395, 2, 509-514 (2012) · Zbl 1261.52005
[450] Sacksteder, R.; Straus, G.; Valentine, FA, A generalization of a theorem of Tietze and Nakajima on local convexity, J. Lond. Math. Soc., 36, 52-56 (1961) · Zbl 0097.31204
[451] Salani, P., Starshapedness of level sets of solutions to elliptic PDEs, Appl. Anal., 84, 12, 1185-1197 (2005) · Zbl 1130.35055
[452] Scheuer, J., Non-scale-invariant inverse curvature flows in hyperbolic space, Calc. Var. Partial Differ. Equ., 53, 1-2, 91-123 (2015) · Zbl 1317.53089
[453] Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. 2nd. expanded edition. Encyclopedia of Mathematics and Its Applications, vol. 151. Cambridge University Press, Cambridge (2014) · Zbl 1287.52001
[454] Schneider, R., Zur einem Problem von Shephard über die Projektionen konvexer Körper, Math. Z., 101, 71-82 (1967) · Zbl 0173.24703
[455] Schu, J., Iterative approximation of fixed points of nonexpansive mappings with starshaped domain, Commentat. Math. Univ. Carol., 31, 2, 277-282 (1990) · Zbl 0717.47022
[456] Schu, J., A fixed point theorem for non-expansive mappings on star-shaped domains, Z. Anal. Anwend., 10, 4, 417-431 (1991) · Zbl 0764.47029
[457] Schu, J., Approximation of fixed points of asymptotically nonexpansive mappings, Proc. Am. Math. Soc., 112, 1, 143-151 (1991) · Zbl 0734.47037
[458] Schuierer, S.; Wood, D., Multiple-guard kernels of simple polygons, J. Geom., 66, 1-2, 161-186 (1999) · Zbl 1005.52001
[459] Schuster, FE, Valuations and Busemann-Petty type problems, Adv. Math., 219, 1, 344-368 (2008) · Zbl 1146.52003
[460] Sengul, U., About the characterization of convex kernel, Int. J. Pure Appl. Math., 19, 2, 269-273 (2005)
[461] Shveidel, A., Separability of starshaped sets and its application to an optimization problem, Optimization, 40, 207-227 (1997) · Zbl 0884.52009
[462] Shveidel, A.: Recession cones of star-shaped and co-star-shaped sets. Optimization and related topics (Ballarat/Melbourne, 1999), pp. 403-414, Appl. Optim., vol. 47. Kluwer Acad. Publ., Dordrecht (2001) · Zbl 1049.90122
[463] Shveidel, A., Star-shapedness and co-star-shapedness of finite unions and intersections of closed half-spaces, Eur. Math. J., 1, 3, 134-147 (2010) · Zbl 1217.52006
[464] Singer, I., Abstract Convex Analysis (1997), Berlin: Wiley, Berlin · Zbl 0898.49001
[465] Singh, SP, An application of a fixed-point theorem to approximation theory, J. Approx. Theory, 25, 1, 89-90 (1979) · Zbl 0399.41032
[466] Sirakov, N.M., Sirakova, N.N.: Inscribing convex polygons in star-shaped objects. In: Combinatorial Image Analysis, pp. 198-211, Lecture Notes in Comput. Sci., vol. 10256. Springer, Cham (2017) · Zbl 1486.52031
[467] Smith, CR, A characterization of star-shaped sets, Am. Math. Monthly, 75, 386 (1968) · Zbl 0161.41603
[468] Smoczyk, K., Starshaped hypersurfaces and the mean curvature flow, Manuscr. Math., 95, 2, 225-236 (1998) · Zbl 0903.53039
[469] Sójka, G., On mappings preserving a family of star bodies, Beitr. Algebra Geom., 44, 155-163 (2003) · Zbl 1036.52011
[470] Sójka, G., Metrics in the family of star bodies, Adv. Math., 13, 117-144 (2013) · Zbl 1269.52008
[471] Soltan, VP, Starshaped sets in the axiomatic theory of convexity, Bull. Acad. Sci. Georgian SSR, 96, 45-48 (1979) · Zbl 0418.52003
[472] Soltan, V.P., Topalè, O.I.: Metric analogues of star-shaped sets (Russian), pp. 122-128, 171, “Shtiinca”, Kishinev (1979) · Zbl 0429.54011
[473] Spiegel, W., Ein Konvergenzsatz für eine gewisse Klasse kompakter Punktmengen, J. Reine Angew. Math., 277, 218-220 (1975) · Zbl 0309.54007
[474] Stanek, JC, A characterization of starshaped sets, Can. J. Math., 29, 4, 673-680 (1977) · Zbl 0357.52009
[475] Stavrakas, N., The dimension of the convex kernel and points of local nonconvexity, Proc. Am. Math. Soc., 34, 222-224 (1972) · Zbl 0235.52004
[476] Stavrakas, N., A generalization of Tietze’s theorem on convex sets in \(\mathbb{R}^3 \), Proc. Am. Math. Soc., 40, 565-567 (1973) · Zbl 0282.52006
[477] Stavrakas, N., A note on starshaped sets, (k)-extreme points and the half ray property, Pac. J. Math., 53, 627-628 (1974) · Zbl 0308.52007
[478] Stavrakas, N., Krasnosel’skiĭ theorems for nonseparating compact sets, Can. Math. Bull., 26, 2, 247-249 (1983) · Zbl 0542.52007
[479] Stavrakas, N., A structure theorem for simply connected \(L_2\) sets, Houst. J. Math., 12, 1, 125-129 (1986) · Zbl 0611.52005
[480] Stavrakas, N., Clear visibility and \(L_2\) sets, Proc. Am. Math. Soc., 103, 4, 1213-1215 (1988) · Zbl 0657.52004
[481] Stavrakas, N., Bounded sets and finite visibility, Topol. Appl., 42, 2, 159-164 (1991) · Zbl 0759.46023
[482] Stavrakas, N., A reduction theorem for the intersection of closed convex hulls, Houst. J. Math., 17, 2, 271-277 (1991) · Zbl 0738.52007
[483] Stavrakas, N.; Jamison, RE, Valentine’s extensions of Tietze’s theorem on convex sets, Proc. Am. Math. Soc., 36, 229-230 (1972) · Zbl 0265.52001
[484] Stečkin, SB, Approximation properties of sets in normed linear spaces, Rev. Math. Pures Appl., 8, 5-18 (1963) · Zbl 0198.16202
[485] Stoddart, AWJ, The shape of level surfaces of harmonic functions in three dimensions, Mich. Math. J., 11, 225-229 (1964) · Zbl 0173.39403
[486] Stoker, JJ, Unbounded convex sets, Am. J. Math., 62, 165-179 (1940) · Zbl 0022.40301
[487] Styer, D., Geometric and annular starlike functions, Complex Variables Theory Appl., 29, 2, 189-191 (1996) · Zbl 0848.30006
[488] Szegö, G., On a certain kind of symmetrization and its applications, Ann. Mat. Pura Appl. (4), 40, 113-119 (1955) · Zbl 0066.40701
[489] Tamássy, L., A characteristic property of the sphere, Pac. J. Math., 29, 439-446 (1969) · Zbl 0181.23702
[490] Taylor, WW, Fixed-point theorems for nonexpansive mappings in linear topological spaces, J. Math. Anal. Appl., 40, 164-173 (1972) · Zbl 0207.45501
[491] Tazawa, Y., A remark on a star-shaped hypersurface with constant reduced mean curvature, J. Fac. Sci. Hokkaido Univ. Ser., I, 21, 122-124 (1970) · Zbl 0206.24102
[492] Thompson, A.C.: Minkowski Geometry. Encyclopedia of Mathematics and its Applications, vol. 63. Cambridge University Press, Cambridge (1996) · Zbl 0868.52001
[493] Tidmore, FE, Extremal structure of star-shaped sets, Pac. J. Math., 29, 461-465 (1969) · Zbl 0176.10703
[494] Tietze, H., Über Konvexheit im kleinen und im grossen und über gewisse den Punkten einer Menge zugeordnete Dimensionszahlen, Math. Z., 28, 697-707 (1928) · JFM 54.0797.01
[495] Todorov, IT; Zidarov, D., Uniqueness of the determination of the shape of an attracting body from the values of its external potential, Dokl. Akad. Nauk SSSR, 120, 262-264 (1958) · Zbl 0090.31601
[496] Topalè, O.I.: Local d-convexity and d-starlike sets. In: Topological Spaces and Algebraic Systems. Mat. Issled. 53, pp. 126-135, 225 (1979) (in Russian) · Zbl 0444.52005
[497] Topalè, O.I.: Extremal points and d-star-shaped sets. In: General Algebra and Discrete Geometry, pp. 108-110, 163, “Shtiintsa”, Kishinev (1980) (in Russian) · Zbl 0541.52004
[498] Topalè, O.I.: The intersection and union of star-shaped sets in a metric space. In: General Algebra and Discrete Geometry, pp. 111-117, 163-164, “Shtiintsa”, Kishinev (1980) (in Russian) · Zbl 0541.52005
[499] Topalè, O.I.: The intersection of maximal star-shaped sets. Izv. Vyssh. Uchebn. Zaved. Mat. 5, 53-54 (1982) translated: Sov. Math. 26(5), 66-68 (1982) (in Russian) · Zbl 0524.52005
[500] Topalè, O.I.: Maximal \(L_n\)-star-shaped sets. Mat. Zametki 32(1), 115-120, 127 (1982) (in Russian) · Zbl 0501.52007
[501] Topalè, O.I.: Some theorems on metric starshapedness of Krasnosel’skiĭ type. In: Investigations in Functional Analysis and Differential Equations, pp. 121-130, 151, “Shtiintsa”, Kishinev (1984) (in Russian) · Zbl 0593.52005
[502] Topalè, O.I.: Finite unions of \(d\)-convex, \(d\)-starshaped and \(L_n\)-starshaped sets. In: Investigations in Numerical Methods and Theoretical Cybernetics, pp. 103-110, 132, “Shtiintsa”, Kishinev (1985) (in Russian)
[503] Topalè, O.I.: A criterion for centrality of a system of maximal starshaped sets. In: Investigations in General Algebra, Geometry, and their Applications (Russian), pp. 138-141, 161, Shtiintsa, Kishinev (1986) (in Russian) · Zbl 0687.52003
[504] Topalè, O.I.: Krasnosel’skiĭ’s theorem for points of local \(d\)-nonconvexity. In: Proc. Sympos. Geom. (Cluj-Napoca and Tîrgu Mureş, 1992), pp. 183-195, Preprint, 93-2, “Babeş-Bolyai” Univ., Cluj-Napoca (1993) (in Russian)
[505] Topalè, O.I., Zarif, A.: A theorem on the union of \(d\)-star-shaped sets. Izv. Akad. Nauk Respub. Moldova Mat. 1, 16-20, 94, 96 (1994) (in Russian) · Zbl 0838.52008
[506] Toranzos, FA, Radial functions of convex and starshaped bodies, Am. Math. Monthly, 74, 278-280 (1967) · Zbl 0145.42802
[507] Toranzos, FA, The dimension of the kernel of a starshaped set, Not. Am. Math. Soc., 14, 832 (1967)
[508] Toranzos, F.A.: Approximation of compact star-shaped sets by special families (Spanish, English summary). Rev. Un. Mat. Argentina 29(1-2), 49-54 (1979-1980) · Zbl 0431.52006
[509] Toranzos, FA, The points of local nonconvexity of starshaped sets, Pac. J. Math., 101, 209-213 (1982) · Zbl 0496.52008
[510] Toranzos, FA, Critical visibility and outward rays, J. Geom., 33, 155-167 (1988) · Zbl 0656.52006
[511] Toranzos, FA, Crowns. A unified approach to starshapedness, Rev. Unión Mat. Argentina, 40, 55-68 (1996) · Zbl 0886.52006
[512] Toranzos, FA; Forte Cunto, A., Clear visibility strikes again, Arch. Math., 58, 307-312 (1992) · Zbl 0758.52005
[513] Toranzos, FA; Forte Cunto, A., Local characterization of starshaped sets, Geom. Dedic., 66, 293-301 (1997)
[514] Toranzos, FA; Forte Cunto, A., Sets expressible as finite unions of starshaped sets, J. Geom., 79, 190-195 (2004) · Zbl 1059.52008
[515] Toranzos, F.A., Nanclares, J.: Convexidad. Cursos, Seminarios y Tesis del PEAM, Venezuela (1978) (in Spanish)
[516] Toranzos, F.A., Zurkowski, V.D.: Perimeter of starshaped plane figures (Spanish. English summary). Math. Notae 29, 95-100 (1981-1982) · Zbl 0522.52007
[517] Tóth, C.D., Toussaint, G.T., Winslow, A.: Open guard edges and edge guards in simple polygons. In: Computational Geometry, Lecture Notes in Comput. Sci., vol. 7579, pp. 54-64. Springer, Cham (2011) · Zbl 1374.68671
[518] Tradacete, P.; Villanueva, I., Continuity and representation of valuations on star bodies, Adv. Math., 329, 361-391 (2018) · Zbl 1400.52015
[519] Treibergs, AE; Wei, SW, Embedded hyperspheres with prescribed mean curvature, J. Differ. Geom., 18, 3, 513-521 (1983) · Zbl 0529.53043
[520] Tsai, D-H, Geometric expansion of starshaped plane curves, Commun. Anal. Geom., 4, 3, 459-480 (1996) · Zbl 0938.35073
[521] Tsai, D.-H.: Expanding embedded plane curves. In: Geometric Evolution Equations, pp. 189-227, Contemp. Math., vol. 367. Amer. Math. Soc., Providence (2005) · Zbl 1075.53065
[522] Tuy, H., Convex Analysis and Global Optimization (1998), Berlin: Kluwer Academic Publishers, Berlin · Zbl 0904.90156
[523] Ubhaya, VA, Generalized isotone optimization with applications to starshaped functions, J. Optim. Theory Appl., 29, 4, 559-571 (1979) · Zbl 0388.65006
[524] Urbas, JIE, On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures, Math. Z., 205, 3, 355-372 (1990) · Zbl 0691.35048
[525] Urrutia, J.: Art gallery and illumination problems. In: Handbook of Computational Geometry, pp. 973-1027. North-Holland, Amsterdam (2000) · Zbl 0941.68138
[526] Valentine, FA, Minimal sets of visibility, Proc. Am. Math. Soc., 4, 917-921 (1953) · Zbl 0052.18103
[527] Valentine, FA, Convex Sets (1964), New York: McGraw-Hill Book Company, New York · Zbl 0129.37203
[528] Valentine, FA, Local convexity and \(L_n\) sets, Proc. Am. Math. Soc., 16, 1305-1310 (1965) · Zbl 0135.40702
[529] Valentine, FA, Two theorems of Krasnosel’skii type, Proc. Am. Math. Soc., 18, 310-314 (1967) · Zbl 0146.44204
[530] Valentine, FA, Visible shorelines, Am. Math. Monthly, 77, 144-152 (1970) · Zbl 0189.52903
[531] Vangeldère, J.: Sur une famille d’ensembles particuliers d’un espace vectoriel (French. English summary). Bull. Soc. Roy. Sci. Liège 38, 158-170 (1969) · Zbl 0187.37102
[532] Van Gompel, G., Defrise, M., Batenburg, K.J.: Reconstruction of a uniform star object from interior \(x\)-ray data: uniqueness, stability and algorithm. Inverse Problems 25(6), 065010, 19 p (2009) · Zbl 1170.65100
[533] Vassiliou, P.J.: Contact geometry of curves. SIGMA Symmetry Integrability Geom. Methods Appl., vol. 5, Paper 098, 27 p (2009) · Zbl 1189.53021
[534] Veselý, L., A simple geometric proof of a theorem for starshaped unions of convex sets, Acta Univ. Carolin. Math. Phys., 49, 2, 79-82 (2008) · Zbl 1190.52003
[535] Viterbo, C., Equivariant Morse theory for starshaped Hamiltonian systems, Trans. Am. Math. Soc., 311, 2, 621-655 (1989) · Zbl 0676.58030
[536] Vrećica, S., A note on starshaped sets, Publ. Inst. Math. (Beograd) (N.S.), 29, 43, 283-288 (1981) · Zbl 0498.52006
[537] Wang, W.; Li, Y., General \(L_p\)-intersection bodies, Taiwan. J. Math., 19, 4, 1247-1259 (2015) · Zbl 1357.52012
[538] Webster, R., Convexity (1994), Oxford: Oxford University Press, Oxford · Zbl 0835.52001
[539] Wu, D.; Zhou, J., The LYZ centroid conjecture for star bodies, Sci. China Math., 61, 7, 1273-1286 (2018) · Zbl 1404.52006
[540] Wu, Z.; Huang, Z.; Wang, W-C; Yang, Y., The direct method of lines for elliptic problems in star-shaped domains, J. Comput. Appl. Math., 327, 350-361 (2018) · Zbl 1372.65320
[541] Xi, D.; Jin, H.; Leng, G., The Orlicz-Brunn-Minkowski inequality, Adv. Math., 260, 350-374 (2014) · Zbl 1357.52004
[542] Xia, Y., Star body valued valuations on \(L^q\)-spaces, Houst. J. Math., 45, 1, 245-265 (2019) · Zbl 1508.46021
[543] Xu, W.; Liu, Y.; Sun, W., On starshaped intuitionistic fuzzy sets, Appl. Math. (Irvine), 2, 1051-1058 (2011)
[544] Yagisita, H., Asymptotic behaviors of star-shaped curves expanding by \(V=1-K\), Differ. Integral Equ., 18, 2, 225-232 (2005) · Zbl 1212.53094
[545] Yaglom, I.M., Boltyanski, V.: Convex Figures. Holt, Rinehart and Winston, New York, 1961 (Russian original: Moscow-Leningrad, 1951)
[546] Yanagi, K., On some fixed point theorems for multivalued mappings, Pac. J. Math., 87, 233-240 (1980) · Zbl 0408.47042
[547] Yaskin, V., The Busemann-Petty problem in hyperbolic and spherical spaces, Adv. Math., 203, 2, 537-553 (2006) · Zbl 1112.52007
[548] Yuan, J., Cheung, W.-S.: \(L_p\) intersection bodies. J. Math. Anal. Appl. 338(2), 1431-1439 (2008) (Corrigendum: J. Math. Anal. Appl. 344 (2008)(1), 592) · Zbl 1136.52006
[549] Zamfirescu, T., Using Baire categories in geometry, Rend. Sem. Mat. Univ. Politec. Torino, 43, 1, 67-88 (1985) · Zbl 0601.52004
[550] Zamfirescu, T., Typical starshaped sets, Aequat. Math., 36, 188-200 (1988) · Zbl 0661.52004
[551] Zamfirescu, T., Description of most starshaped sets, Math. Proc. Camb. Phil. Soc., 106, 245-251 (1989) · Zbl 0739.52011
[552] Zamfirescu, T., Baire categories in convexity, Atti Sem. Mat. Fis. Univ. Modena, 39, 1, 139-164 (1991) · Zbl 0780.52003
[553] Zhang, D., \(L_p\)-mixed intersection bodies, Math. Inequal. Appl., 19, 425-438 (2016) · Zbl 1341.52011
[554] Zhang, D.; Yang, Y., The dual generalized Chernoff inequality for star-shaped curves, Turk. J. Math., 40, 2, 272-282 (2016) · Zbl 1424.52007
[555] Zhang, GY, Centered bodies and dual mixed volumes, Trans. Am. Math. Soc., 345, 2, 777-801 (1994) · Zbl 0812.52005
[556] Zhang, GY, Intersection bodies and the Busemann-Petty inequalities in \(\mathbb{R}^4 \), Ann. Math. (2), 140, 2, 331-346 (1994) · Zbl 0826.52011
[557] Zhang, GY, Intersection bodies and polytopes, Mathematika, 46, 1, 29-34 (1999) · Zbl 0964.52007
[558] Zhang, L.; Xia, Z.; Gao, Y.; Wang, M., Star-kernels and star-differentials in quasidifferential analysis, J. Convex Anal., 9, 1, 139-158 (2002) · Zbl 1136.49306
[559] Zhang, S., Star-shaped sets and fixed points of multivalued mappings, Math. Jpn., 36, 2, 327-334 (1991) · Zbl 0752.47017
[560] Zhao, C-J, On intersection and mixed intersection bodies, Geom. Dedic., 141, 109-122 (2009) · Zbl 1175.52011
[561] Zhao, C-J, Orlicz dual mixed volumes, Results Math., 68, 1-2, 93-104 (2015) · Zbl 1329.52008
[562] Zhu, B.; Zhou, J.; Xu, W., Dual Orlicz-Brunn-Minkowski theory, Adv. Math., 264, 700-725 (2014) · Zbl 1307.52004
[563] Zhu, G., The Orlicz centroid inequality for star bodies, Adv. Appl. Math., 48, 2, 432-445 (2012) · Zbl 1271.52008
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