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Approximation of convex sets by polytopes. (English. Russian original) Zbl 1161.52001

J. Math. Sci., New York 153, No. 6, 727-762 (2008); translation from Sovrem. Mat., Fundam. Napravl. 22, 5-37 (2007).
This survey paper updates the well-known surveys of P. M. Gruber [Aspects of approximation of convex bodies. Handbook of convex geometry. Volume A. Amsterdam: North-Holland. 319–345 (1993; Zbl 0791.52007) and Approximation by convex polytopes. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 440, 173–203 (1994; Zbl 0824.52007)] on different aspects of polyhedral approximation of convex bodies and some adjacent problems. A list of 230 references is given. Nearly half of the papers cited in the survey have appeared in the last ten years.

MSC:

52-02 Research exposition (monographs, survey articles) pertaining to convex and discrete geometry
52A27 Approximation by convex sets
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References:

[1] F. Affentranger and R. Schneider, ”Random projections of regular simplexes,” Discr. Comput. Geom., 7, 219–226 (1992). · Zbl 0751.52002 · doi:10.1007/BF02187839
[2] F. Affentranger and J. Wieaker, ”On the convex hull of uniform random points in a simple d-polytope,” Discr. Comput. Geom., 6(1991), 291–305 (1991). · Zbl 0725.52004 · doi:10.1007/BF02574691
[3] H.-K. Ahn, P. Brass, O. Cheong, H.-S. Na, C.-S. Shin, and A. Vigneron, ”Approximation algorithms for inscribing or circumscribing an axially symmetric polygon to a convex polygon,” in: Computing and Combinatorics. Proc. 10th Annual Int. Conf. COCOON 2004, Jeju Island, Korea, August 17–20, 2004, Lect. Notes Comput. Sci., 3106, Springer-Verlag, Berlin (2004), pp. 259–267. · Zbl 1091.68109
[4] G. E. Andrews, ”A lower bound for the volume of strictly convex bodies with many boundary lattice points,” Trans. Amer. Math. Soc., 106, 270–279 (1963). · Zbl 0118.28301 · doi:10.1090/S0002-9947-1963-0143105-7
[5] V. I. Arnold, ”Statistics of integer-valued convex polygons,” Funkts. Anal. Prilozh., 14, 1–3 (1980).
[6] E. Asplund, ”Comparison of plane symmetric convex bodies and parallelograms,” Math. Scand., 8, 171–180 (1960). · Zbl 0104.17001
[7] M. J. Atallah, ”A linear time algorithm for the Hausdorff distance between convex polygons,” Inf. Process. Lett., 17, 207–209 (1983). · Zbl 0527.68051 · doi:10.1016/0020-0190(83)90042-X
[8] M. J. Atallah, C. C. Ribeiro, and S. Lifschitz, ”A linear time algorithm for the computation of some distance functions between convex polygons,” RAIRO, Rech. Oper., 25, 413–424 (1991). · Zbl 0770.68109
[9] I. K. Babenko, ”Asymptotic volume of tori and geometry of convex bodies,” Mat. Zametki, 44, 177–190 (1988). · Zbl 0653.52008
[10] A. Balog and I. Bárány, ”On the convex hull of the integer points in a disc,” in: Discrete and Computational Geometry. Proc. DIMACS Spec. Year Workshops 1989–90, DIMACS, Ser. Discr. Math. Theor. Comput. Sci., 6, 39–44 (1991).
[11] A. Balog and I. Bárány, ”On the convex hull of the integer points in a disk,” Discr. Comput. Geom., 6, 39–44 (1992). · Zbl 0746.52017
[12] A. Balog and J.-M. Deshorlies, ”On some convex lattice polytopes,” in: Number Theory in Progress, de Gruyter (1999), pp. 591–606. · Zbl 0931.52005
[13] Yu. M. Baryshnikov and R. A. Vitale, ”Regular simplices and Gaussian samples,” Discr. Comput. Geom., 11, 141–147 (1994). · Zbl 0795.52002 · doi:10.1007/BF02574000
[14] I. Bárány, ”Random polytopes in smooth convex bodies,” Mathematica, 39, 81–92 (1992). · Zbl 0765.52009
[15] I. Bárány, ”The limit shape of convex lattice polygons,” Discr. Comput. Geom., 13, 270–295 (1995). · Zbl 0824.52001
[16] I. Bárány, ”Affine perimeter and limit shape,” J. Reine Angew. Math., 484, 71–84 (1997). · Zbl 0864.52010 · doi:10.1515/crll.1997.484.71
[17] I. Bárány, ”Sylvester’s question: The probability that n points are in convex position,” Ann. Probab., 27, 2020–2034 (1999). · Zbl 0959.60006 · doi:10.1214/aop/1022677559
[18] I. Bárány, ”A note on Sylvester’s four-point problem,” Stud. Sci. Math. Hung., 38, 73–77 (2001). · Zbl 1002.60008
[19] I. Bárány, ”Random points, convex bodies, lattices,” in: Proc. Int. Congr. Math. 2002, 3 (2002), pp. 527–535. · Zbl 1010.52007
[20] I. Bárány and K. Böröczky, Jr., ”Lattice points on the boundary of the integer hull,” in: Discrete Geometry. In honor of W. Kuperberg’s 60th birthday, Pure Appl. Math., Marcel Dekker, 253, (2003), pp. 33–47.
[21] I. Bárány and C. Buchta, ”Random polytopes in a convex polytope, independence of shape, and concentration of vertices,” Math. Ann., 297, 467–497 (1993). · Zbl 0788.52003 · doi:10.1007/BF01459511
[22] I. Bárány and L. Dalla, ”Few points to generate a random polytope,” Mathematika, 44, 325–331 (1997). · Zbl 0902.52002 · doi:10.1112/S0025579300012638
[23] I. Bárány and Z. Füredi, ”Approximation of the sphere by polytopes having few vertices,” Proc. Am. Math. Soc., 102, 651–659 (1988). · Zbl 0669.52003 · doi:10.1090/S0002-9939-1988-0928998-8
[24] I. Bárány and Z. Füredi, ”On the shape of the convex hull of random points,” Probab. Theory Rel. Fields, 77, 231–240 (1988). · Zbl 0639.60015 · doi:10.1007/BF00334039
[25] I. Bárány, R. Howe, and L. Lovász, ”On integer points in polyhedra: A lower bound,” Combinatorica, 12, 135–142 (1992). · Zbl 0754.52005 · doi:10.1007/BF01204716
[26] I. Bárány and D. G. Larman, ”The convex hull of the integer points in the large ball,” Math. Ann., 312, 167–181 (1998). · Zbl 0927.52020 · doi:10.1007/s002080050217
[27] I. Bárány and D. G. Larman, ”Convex bodies, economic cap coverings, random polytopes,” Mathematika, 35, 274–291 (1988). · Zbl 0674.52003 · doi:10.1112/S0025579300015266
[28] I. Bárány and J. Matousek, ”The randomized integer convex hull,” Discr. Comput. Geom., 33, 3–25 (2005). · Zbl 1078.52005 · doi:10.1007/s00454-003-0836-1
[29] I. Bárány and A. Pór, ”On 0-1 polytopes with many faces,” Adv. Math., 161, 209–228 (2001). · Zbl 0988.52014 · doi:10.1006/aima.2001.1991
[30] I. Bárány, G. Rote, W. Steiger, and C.-H. Zhang, ”A central limit theorem for convex chains in the square,” Discr. Comput. Geom., 23, 35–50 (2000). · Zbl 0954.52008 · doi:10.1007/PL00009490
[31] I. Bárány and N. Tokushige, ”The minimum area of convex lattice n-gons,” Combinatorica, 24, 171–185 (2004). · Zbl 1068.52022 · doi:10.1007/s00493-004-0012-0
[32] I. Bárány and A. M. Vershik, ”On the number of convex lattice polytopes,” GAFA J., 2, 381–393 (1992). · Zbl 0772.52010 · doi:10.1007/BF01896660
[33] A. Bielecki and K. Radziszewski, ”Sur les parallélépipèdes inscrits dans les corps convexes,” Ann. Univ. M. Curie-Sclodowska, Sec. A, 7, 97–100 (1954). · Zbl 0071.38002
[34] W. Blaschke, Kreis und Kugel, Walter de Gruyter, Berlin (1956).
[35] W. Blaschke, ”Über affine Geometrie, III,” Ber. Verh. Sächs. Ges. Wiss. Leipzig, Math.-Phys., 69, 3–12 (1917).
[36] W. Blaschke, ”Über affine Geometrie, IX,” Ber. Verh. Sächs. Ges. Wiss. Leipzig, Math.-Phys., 69, 412–420 (1917).
[37] W. Blaschke, Affine Differenzialgeometrie, Springer-Verlag, 1923.
[38] J. Bokowski, ”Konvexe Körper approximierande Polytopclassen,” Elem. Math., 32, 88–90 (1977). · Zbl 0359.52005
[39] J. Bokowski and C. Schulz, ”Dichte Klassen konvexer Polytope,” Math. Z., 160, 173–182 (1978). · Zbl 0369.52010 · doi:10.1007/BF01214267
[40] J. Bokowski and P. Mani-Levinska, ”Approximation of convex bodies by polytopes with uniformly bounded valences,” Monatsh. Math., 104, 261–264 (1987). · Zbl 0627.52003 · doi:10.1007/BF01294649
[41] W. Boratyński, ”Approximation of convex bodies by parallelotopes,” in: Proc. 7th Int. Conf. on Engineering, Computer Graphics, and Descriptive Geometry, Cracow, Poland (1996), pp. 259–260. · Zbl 0924.52003
[42] K. Böröczky, Jr., ”Approximation of general smooth convex bodies,” Adv. Math, 153, 325–341 (2000). · Zbl 0964.52010 · doi:10.1006/aima.1999.1904
[43] K. Böröczky, Jr., ”Polytopal approximation bounding the number of k-faces,” J. Approx. Theory, 102, 263–285 (2000). · Zbl 0974.52006 · doi:10.1006/jath.1999.3413
[44] K. Böröczky, Jr., ”The error of polytopal approximation with respect to the symmetric difference metric and the L p metric,” Isr. J. Math, 117, 1–28 (2000). · Zbl 0986.52004 · doi:10.1007/BF02773561
[45] K. Böröczky, Jr., ”About the error term for best approximation with respect to the Hausdorff related metrics,” Discr. Comput. Geom., 25, 293–309 (2001). · Zbl 0987.52004 · doi:10.1007/s004540010088
[46] K. Böröczky, Jr. and M. Ludwig, ”Approximation of convex bodies and a momentum lemma for power diagram,” Monatsh. Math., 127, 101–110 (1999). · Zbl 0933.52023 · doi:10.1007/s006050050026
[47] K. Böröczky, Jr. and M. Reitzner, ”Approximation of smooth convex bodies by random circumscribed polytopes,” Ann. Appl. Probab., 14, 239–273 (2004). · Zbl 1049.60009 · doi:10.1214/aoap/1075828053
[48] J. Bourgain and J. Lindenstrauss, ”Distribution of points on sphere and approximation by zonotopes,” Isr. J. Math., 64, 25–31 (1988). · Zbl 0667.52001 · doi:10.1007/BF02767366
[49] J. Bourgain, J. Lindenstrauss, and V. D. Milman, ”Approximation of zonoids by zonotopes,” Acta Math., 162, 73–141 (1989). · Zbl 0682.46008 · doi:10.1007/BF02392835
[50] E. M. Bronshtein, ”Extremal convex functions,” Sib. Mat. Zh., 19, 10–18 (1978). · Zbl 0411.52001 · doi:10.1007/BF00967357
[51] E. M. Bronshtein and L. D. Ivanov, ”Approximation of convex sets by polyhedrons,” Sib. Mat. Zh., 16, 1110–1112 (1975). · Zbl 0325.52015
[52] C. Buchta, ”Über die konvexe Hölle von Zufallspunkten in Eibereichen,” Elem. Math., 38, 153–156 (1983). · Zbl 0521.52005
[53] C. Buchta, ”Stochastische Approximation konvexer Polygone,” Z. Wahrscheinlichkeitstheor. Verw. Geb., 67, 283–304 (1984). · Zbl 0536.60020 · doi:10.1007/BF00535006
[54] C. Buchta, ”A note on the volume of a random polytope in a tetrahedron,” Ill. J. Math., 30, 653–659 (1986). · Zbl 0585.52001
[55] C. Buchta, ”A remark on random approximation of simple polytopes,” Anz. Österr. Akad. Wiss., Math.-Naturwiss., 2, 17–20 (1989). · Zbl 0683.52006
[56] C. Buchta and M. Reitzner, ”What is the expected volume of a tetrahedron whose vertices are chosen at random from a given tetrahedron?” Anz. Österr. Akad. Wiss., Math.-Naturwiss., 8, 63–68 (1992). · Zbl 0774.60016
[57] C. Buchta and M. Reitzner, ”Equiaffine inner parallel curves of a plane convex body and the convex hulls of randomly chosen points,” Probab. Theory Relat. Fields, 108, 385–415 (1997). · Zbl 0882.52004 · doi:10.1007/s004400050114
[58] R. E. Burkard, H. W. Hamacher, and G. Rote, ”Sandwich approximation of univariate convex functions with its application to separable convex programming,” Naval Res. Logist., 38, 911–924 (1991). · Zbl 0755.90066
[59] L. V. Burmistrova, On a new method of approximation of convex compact bodies by polyhedrons, Preprint, Comput. Center of the Russian Academy of Sciences (1999). · Zbl 0938.52007
[60] L. V. Burmistrova, ”On a new method of approximation of convex compact bodies by polyhedrons,” Zh. Vychisl. Mat. Mat. Fiz., 40, 1475–1490 (2000).
[61] V. A. Bushenkov, ”Iterative method of construction of optimal projections of convex polyhedron sets,” Zh. Vychisl. Mat. Mat. Fiz., 25, 1285–1292 (1985). · Zbl 0588.90057
[62] V. A. Bushenkov and A. V. Lotov, ”Constructions and applications of generalized attainability sets,” Preprint, Comput. Center of the Russian Academy of Sciences (1982). · Zbl 0478.49029
[63] A. J. Cabo and P. Groeneboom, ”Limit theorems for functionals of convex hulls,” Probab. Theory Relat. Fields, 100, 31–55 (1994). · Zbl 0808.60019 · doi:10.1007/BF01204952
[64] G. H. L. Cheang and A. R. Barron, ”A better approximation for balls,” J. Approx. Theory, 104, 183–203 (2000). · Zbl 0991.41012 · doi:10.1006/jath.1999.3441
[65] L. Chen, ”New analysis of the sphere covering problems and optimal polytope approximation of convex bodies,” J. Approx. Theory, 133, 134–145 (2005). · Zbl 1072.65021 · doi:10.1016/j.jat.2004.12.009
[66] O. L. Chernykh, ”On the construction of the convex span of a finite set of points in approximate calculations,” Zh. Vychisl. Mat. Mat. Fiz., 28, 1386–1396 (1988).
[67] K. L. Clarkson, ”Algorithms for polytope covering and approximation,” in: Proc. 3rd Workshop on Algorithms and Data Structures, Lect. Notes Comput. Sci., 709, Springer-Verlag (1993), pp. 246–252.
[68] W. Cook, M. Hartmann, R. Kannan, and C. McDiarmid, ”On integer points in polyhedra,” Combinatorica, 12, 27–37 (1992). · Zbl 0757.52013 · doi:10.1007/BF01191202
[69] L. Dalla and D. G. Larman, ”Volumes of a random polytope in a convex set,” in: Applied Geometry and Discrete Mathematics, Festschr. 65th Birthday Victor Klee, DIMACS, Ser. Discret. Math. Theor. Comput. Sci., 4 (1991), pp. 175–180. · Zbl 0751.52003
[70] L. Devroye, ”On the oscillation of the expected number of extreme points of a random set,” Stat. Probab. Lett., 11, 281–286 (1991). · Zbl 0723.60016 · doi:10.1016/0167-7152(91)90036-Q
[71] P. C. Doyle, J. C. Lagarian, and D. Randall, ”Self-parking of centrally symmetric convex bodies in \(\mathbb{R}\)2,” Discr. Comput. Geom., 8, 171–189 (1992). · Zbl 0756.52016 · doi:10.1007/BF02293042
[72] A. Dvoretzky, ”Some results on convex bodies and Banach spaces,” in: Proc. Int. Symp. Linear Spaces, Jerusalem, 1960, Jerusalem Academic Press; Pergamon, Oxford (1961), pp. 123–160.
[73] A. Dvoretzky and C. A. Rogers, ”Absolute and unconditional convergence in normed linear spaces,” Proc. Nac. Acad. Sci. USA, 36, 192–197 (1950). · Zbl 0036.36303 · doi:10.1073/pnas.36.3.192
[74] R. M. Dudley, ”Metric entropy of some classes of sets with differentiable boundaries,” J. Approx. Theory, 10, 227–236 (1974). · Zbl 0275.41011 · doi:10.1016/0021-9045(74)90120-8
[75] R. Dwyer, ”On the convex hull of random points in a polytope,” J. Appl. Probab., 25, 688–699 (1988). · Zbl 0672.60019 · doi:10.2307/3214289
[76] R. Dwyer and R. Kannan, ”Convex hull of randomly chosen points from a polytope. Parallel algorithms and architectures,” in: Proc. Int. Workshop, Suhl/GDR 1987, Math. Res., 38, 16–24 (1987). · Zbl 0636.52007
[77] M. E. Dyer and A. M. Frieze, ”On the complexity of computing the volume of a polyhedron,” SIAM J. Comput., 17, 967–974 (1988). · Zbl 0668.68049 · doi:10.1137/0217060
[78] M. Dyer and A. Frieze, ”Computing the volume of convex bodies: A case where randomness provably helps,” in: Probabilistic Combinatorics and Its Applications, Short Course, Proc. Symp. Appl. Math., 44 (1991), pp 123–169. · Zbl 0754.68052
[79] M. E. Dyer, A. Frieze, and R. Kannan, ”A random polynomial-time algorithm for approximating the volume of convex bodies,” J. Assoc. Comput. Mach., 38, 1–17 (1991). · Zbl 0799.68107
[80] M. E. Dyer, Z. Fúredi, and C. McDiarmid, ”Random volumes in the n-cube,” in: Polyhedral Combinatorics, Proc. Workshop, Morristown/NJ (USA) 1989, DIMACS, Ser. Discrete Math. Theor. Comput. Sci., 1 (1990), pp. 33–38. · Zbl 0736.52002
[81] M. E. Dyer, Z. Fúredi, and C. McDiarmid, ”Volumes spanned by random points in the hypercube,” Random Struct. Algorithms, 3, 91–106 (1992). · Zbl 0755.60013 · doi:10.1002/rsa.3240030107
[82] M. Dyer, P. Gritzmann, and A. Hufnagel, ”On the complexity of computing mixed volumes,” SIAM J. Comput., 27, 356–400 (1998). · Zbl 0909.68193 · doi:10.1137/S0097539794278384
[83] S. M. Dzholdybaeva and G. K. Kamenev, ”Numerical study of the effectiveness of the algorithm of approximation of convex bodies by polyhedrons,” Zh. Vychisl. Mat. Mat. Fiz., 32, 857–866 (1992). · Zbl 0779.65098
[84] R. V. Efremov, ”Estimate of the effectiveness of the adaptive approximation algorithm for smooth convex bodies in the two-dimensional case,” Vestn. Mosk. Univ., Ser. 15, 2, 29–32 (2000). · Zbl 0999.34005
[85] R. V. Efremov and G. K. Kamenev, ”A priori estimate of the asymptotic effectiveness of one class of polyhedral approximation algorithms for convex bodies,” Zh. Vychisl. Mat. Mat. Fiz., 42, 23–32 (2002). · Zbl 1057.52001
[86] R. V. Efremov, G. K. Kamenev, and A. V. Lotov, ”On the construction of efficient descriptions of polyhedrons based on the duality theory for convex sets,” Dokl. Ross. Akad. Nauk, 399, 594–596 (2004).
[87] H. G. Eggleston, ”Approximation of plane convex curves, I. Dowker-type theorems,” Proc. London Math. Soc., 7, No. 3, 351–377 (1957). · Zbl 0080.15603 · doi:10.1112/plms/s3-7.1.351
[88] H. G. Eggleston, Convexity, Cambridge (1958).
[89] T. Esterman, ”Über der Vektorenbereich eines konvexen Körpers,” Math. Z., 28, 471–475 (1928). · JFM 54.0798.03 · doi:10.1007/BF01181177
[90] E. Fabińska and M. Lassak, ”Large equilateral triangles inscribed in the unit disk of a Minkowski plane,” Contrib. Algebra Geom., 45, 517–525 (2004). · Zbl 1077.52007
[91] U. Faigle, N. Gademann, and W. Kern, ”A random polynomial time algorithm for well-routing convex bodies,” Discr. Appl. Math., 58, 117–144 (1995). · Zbl 0818.68094 · doi:10.1016/0166-218X(93)E0123-G
[92] L. Fejes Tóth, ”Approximation by polygons and polyhedra,” Bull. Amer. Math. Soc., 54, 431–438 (1948). · Zbl 0031.27806 · doi:10.1090/S0002-9904-1948-09022-X
[93] L. Fejes Tóth, Lagerungen in der Ebene auf der Kugel und im Raum, Springer-Verlag, Berlin-Göttingen-Heidelberg (1953). · Zbl 0052.18401
[94] L. Fejes T’oth, ”Approximation of convex domains by polygons,” Stud. Sci. Math. Hung., 15, 133–138 (1980). · Zbl 0514.52008
[95] P. Fischer, ”Sequential and parallel algorithms for finding a maximum convex polygon,” Comput. Geom., 7, 187–200 (1997). · Zbl 0869.68114 · doi:10.1016/0925-7721(95)00035-6
[96] S. Finch and I. Hueter, ”Random convex hulls: A variance revisited,” Adv. Appl. Probab., 36, 981–986 (2004). · Zbl 1068.60020 · doi:10.1239/aap/1103662954
[97] T. Fleiner, V. Kaibel, and G. Rote, ”Upper bounds on the maximal number of facets of 0/1-polytopes,” Eur. J. Combinat., 21, 121–130 (2000). · Zbl 0951.52007 · doi:10.1006/eujc.1999.0326
[98] A. Florian, ”On the perimeter deviation of a convex disk from a polygon,” Rend. Ist. Mat. Univ. Trieste, 24, 177–191 (1992). · Zbl 0819.52003
[99] A. Florian, ”Approximation of convex discs by polygons: The perimeter deviation,” Stud. Sci. Math. Hung., 27, 97–118 (1992). · Zbl 0801.52004
[100] E. Floyd, ”Real-valued mappings of spheres,” Proc. Amer. Math. Soc., 6, 957–959 (1955). · Zbl 0067.40702 · doi:10.1090/S0002-9939-1955-0073978-9
[101] B. Fruhwirth, R. E. Burkard, and G. Rote, ”Approximation of convex curves with applications to the minimum cost flow problem,” Eur. J. Oper. Res., 42, 326–338 (1989). · Zbl 0684.65069 · doi:10.1016/0377-2217(89)90443-8
[102] C. M. Fulton and S. K. Stein, ”Parallelograms inscribed in convex curves,” Amer. Math. Monthly, 67, 257–258 (1960). · Zbl 0090.12703 · doi:10.2307/2309688
[103] P. Georgiev, in: Proc 13 Bulgarian Conf. Math. (1984), pp. 289–303.
[104] S. Glasauer and P. M. Gruber, ”Asymptotic estimates for best and stepwise approximation of convex bodies, III,” Forum Math., 9, 383–404 (1997). · Zbl 0889.52006 · doi:10.1515/form.1997.9.383
[105] S. Glasauer and R. Schneider, ”Asymptotic approximation of smooth convex bodies by polytopes,” Forum Math., 8, 363–377 (1996). · Zbl 0855.52001 · doi:10.1515/form.1996.8.363
[106] A. M. Gleason, ”A curvature formula,” Amer. J. Math., 101, 86–93 (1979). · Zbl 0423.53002 · doi:10.2307/2373940
[107] E. D. Gluskin, ”On the sum of intervals,” in: Geometric Aspects of Functional Analysis. Proc. Israel Semin. (GAFA) 2001–2002, Lect. Notes Math., 1807, Springer-Verlag, Berlin (2003), pp. 122–130.
[108] Y. Gordon, M. Meyer, and S. Reisner, ”Volume approximation of convex bodies by polytopes: A constructive method,” Stud. Math., 111, 81–95 (1994). · Zbl 0808.52001
[109] Y. Gordon, M. Meyer, and S. Reisner, ”Constructing a polytope to approximate a convex body,” Geom. Dedic., 57, 217–222 (1995). · Zbl 0838.52003 · doi:10.1007/BF01264939
[110] Y. Gordon, S. Reisner, and C. Schütt, ”Umbrellas and polytopal approximation of the Euclidean ball,” J. Approx. Theory, 90, 9–22 (1997). · Zbl 0885.52006 · doi:10.1006/jath.1996.3065
[111] P. Gritzmann, V. Klee, and D. Larman, ”Largest j-simplices in d-polytopes,” Discr. Comput. Geom., 13, 477–515 (1995). · Zbl 0826.52014 · doi:10.1007/BF02574058
[112] P. Gritzman and M. Lassak, ”Estimates for the minimal wight of polytopes inscribed in convex bodies,” Discr. Comput. Geom., 4, 627–635 (1989). · Zbl 0692.52002 · doi:10.1007/BF02187752
[113] P. Groeneboom, ”Limit theorems for convex hulls,” Probab. Theory Relat. Fields, 79, 327–368 (1988). · Zbl 0635.60012 · doi:10.1007/BF00342231
[114] W. Gross, ”Über affine Geometrie, XIII,” Ber. Verh. Sächs. Acad. Wis. Leipzig. Math.-Nat, 70, 38–54 (1918).
[115] P. M. Gruber, ”In most cases approximation is irregular,” Rend. Sem. Mat. Univers. Torino, 41, 19–33 (1983). · Zbl 0562.41030
[116] P. M. Gruber, ”Volume approximation of convex bodies by inscribed polytopes,” Math Ann., 281, 229–245 (1988). · Zbl 0628.52006 · doi:10.1007/BF01458430
[117] P. M. Gruber, ”Volume approximation of convex bodies by circumscribed polytopes,” in: Applied Geometry and Discrete Mathematics, DIMACS Ser. 4, Amer. Math. Soc., Providence (1991), pp. 309–317. · Zbl 0739.52004
[118] P. M. Gruber, ”The space of convex bodies,” in: Handbook of Convex Geometry, Elsevier (1993), pp. 301–317. · Zbl 0791.52004
[119] P. M. Gruber, ”Aspects of approximation of convex bodies,” in: Handbook of Convex Geometry, Elsevier (1993), pp. 319–345. · Zbl 0791.52007
[120] P. M. Gruber, ”Asymptotic estimates for best and stepwise approximation of convex bodies, I,” Forum Math., 5, 281–297 (1993). · Zbl 0780.52005 · doi:10.1515/form.1993.5.281
[121] P. M. Gruber, ”Asymptotic estimates for best and stepwise approximation of convex bodies, II,” Forum Math., 5, 521–537 (1993). · Zbl 0788.41020 · doi:10.1515/form.1993.5.521
[122] P. M. Gruber, ”Approximation by convex polytopes. Polytopes: Abstract, convex and computational,” in: Proc. NATO Adv. Stud. Inst. Scarborough, Ontario, Canada, August 20–September 3, 1993, NATO ASI Ser., Ser. C, Math. Phys. Sci., 440, Kluwer Academic Publishers, Dordrecht (1994), pp. 173–203. · Zbl 0824.52007
[123] P. M. Gruber, ”Comparison of best and random approximation of convex bodies by polytopes,” Rend. Circ. Math. Palermo (2), 50, 189–216 (1997). · Zbl 0896.52014
[124] P. M. Gruber, ”A short analytic proof of Fejes Tóth’s theorem on sums of moments,” Aequationes Math., 58, 291–295 (1999). · Zbl 1006.52015 · doi:10.1007/s000100050116
[125] P. M. Gruber, ”Error of asymptotic formulae for volume approximation of convex bodies in E 3,” Tr. Mat. Inst. Steklova, 239, 106–117 (2002). · Zbl 1068.52007
[126] P. M. Gruber, ”Error of asymptotic formulae for volume approximation of convex bodies in E 3,” Monatsh. Math., 135, 271–304 (2002). · Zbl 1006.52002
[127] P. M. Gruber and P. Kenderov, ”Approximation of convex bodies by polytopes,” Rend. Circ. Mat. Palermo (2), 31, 195–225 (1982). · Zbl 0494.52003 · doi:10.1007/BF02844354
[128] B. Grünbaum, Studies in Combinatorial Geometry and the Theory of Convex Bodies [Russian translation], Nauka, Moscow (1971).
[129] Q. Guo and S. Kajiser, ”Approximation of convex bodies by convex bodies,” Northeast Math. J, 19, 323–332 (2003). · Zbl 1071.52010
[130] H. Hadwiger, ”Volumschätzung für die eine Eikörper überdeckenden und unterdeckenden Parallelotope,” Elem. Math., 10, 122–124 (1955). · Zbl 0066.40605
[131] H. Hadwiger, Lectures on the Volume, Surface Area, and Isoperimetry [Russian translation], Nauka, Moscow (1966).
[132] A. C. Hayes and D. G. Larman, ”The vertices of the knapsack polytope,” Discr. Appl. Math., 6, 135–138 (1983). · Zbl 0523.90063 · doi:10.1016/0166-218X(83)90067-7
[133] I. Hueter, ”The convex hull of a normal sample,” Adv. Appl. Probab., 26, 855–875 (1994). · Zbl 0815.60013 · doi:10.2307/1427894
[134] I. Hueter, ”Limit theorems for the convex hull of random points in higher dimensions,” Trans. Amer. Math. Soc., 351, 4337–4363 (1999). · Zbl 0944.60018 · doi:10.1090/S0002-9947-99-02499-X
[135] D. Hug, G. O. Munsonius, and M. Reitzner, ”Asymptotic mean values of Gaussian polytopes,” Beitr. Algebra Geom., 45, 531–548 (2004). · Zbl 1082.52003
[136] D. Hug and M. Reitzner, ”Gaussian polytopes: Variances and limit theorems,” Adv. Appl. Probab., 37, 297–320 (2005). · Zbl 1089.52003 · doi:10.1239/aap/1118858627
[137] S. Johansen, ”The extremal convex functions,” Math. Scand., 34, 61–68 (1974). · Zbl 0286.26002
[138] S. Kakutani, ”A proof that there exists a circumscribing cube around any bounded closed convex sets in R 3,” Ann. Math., 43, 739–741 (1942). · Zbl 0061.38309 · doi:10.2307/1968964
[139] G. K. Kamenev, ”On one class of adaptive algorithms of approximation of convex bodies by polyhedrons,” Zh. Vychisl. Mat. Mat. Fiz., 32, 136–152 (1992). · Zbl 0756.52006
[140] G. K. Kamenev, ”On the effectiveness of Hausdorff algorithms of polyhedral approximations of convex bodies,” Zh. Vychisl. Mat. Mat. Fiz., 33, 796–805 (1993). · Zbl 0804.52003
[141] G. K. Kamenev, ”On the examining of one algorithm of approximation of convex bodies,” Zh. Vychisl. Mat. Mat. Fiz., 34, No. 4, 521–528 (1994).
[142] G. K. Kamenev, ”Algorithm of approaching polyhedrons,” Zh. Vychisl. Mat. Mat. Fiz., 36, 134–147 (1996). · Zbl 1161.52301
[143] G. K. Kamenev, ”Effective algorithms of internal polyhedral approximation of nonsmooth convex bodies,” Zh. Vychisl. Mat. Mat. Fiz., 39, 423–427 (1999). · Zbl 0970.65019
[144] G. K. Kamenev, ”On approximative properties of nonsmooth convex disks,” Zh. Vychisl. Mat. Mat. Fiz., 40, 1464–1474 (2000). · Zbl 1007.52003
[145] G. K. Kamenev, ”On a method of polyhedral approximation of convex bodies that is optimal with respect to the number of calculations of support and distance functions,” Dokl. Ross. Akad. Nauk, 388, 309–311 (2003). · Zbl 1038.65511
[146] G. K. Kamenev, ”Self-dual adaptive polyhedral approximation algorithms for convex bodies,” Zh. Vychisl. Mat. Mat. Fiz., 43, 1123–1137 (2003). · Zbl 1075.52505
[147] R. Kannan, L. Lovsz, and M. Simonovits, ”Random walks and an O*(n 5) volume algorithm for convex bodies,” Random Struct. Algorithms, 11, 1–50 (1997). · Zbl 0895.60075 · doi:10.1002/(SICI)1098-2418(199708)11:1<1::AID-RSA1>3.0.CO;2-X
[148] B. S. Kashin, ”On parallelepipeds of minimal volume that contain a convex body,” Mat. Zametki, 45, 134–135 (1989). · Zbl 0663.52002
[149] P. Kenderov, ”Approximation of plane convex compacta by polygons,” Dokl. Bulg. Akad. Nauk, 33, 889–891 (1980). · Zbl 0446.52013
[150] P. Kenderov, ”Polygonal approximation of plane convex compacta,” J. Approx. Theory, 38, 221–239 (1983). · Zbl 0521.41021 · doi:10.1016/0021-9045(83)90130-2
[151] K.-H. Küfer, ”On the approximation of a ball by random polytopes,” Adv. Appl. Probab., 26, 876–892 (1994). · Zbl 0812.60017 · doi:10.2307/1427895
[152] V. Klee, ”Facet-centroids and volume minimization,” Stud. Sci. Math. Hungar., 21, 143–147 (1986). · Zbl 0547.52002
[153] V. Klee and M. C. Laskowski, ”Finding the smallest triangle containing a given convex polygon,” J. Algorithms, 6, 359–366 (1985). · Zbl 0577.52003 · doi:10.1016/0196-6774(85)90005-7
[154] M. Kochol, ”A note on approximation of a ball by polytopes,” Discr. Optim., 1, 229–231 (2004). · Zbl 1085.52502 · doi:10.1016/j.disopt.2004.07.003
[155] A. Kosinński, ”A proof of an Auerbach-Banach-Mazur-Ulam theorem on convex bodies,” Colloq. Math., 4, 216–218 (1957). · Zbl 0083.38401
[156] D. Kylzow, A. Kuba, and A. Volcic, ”An algorithm for reconstructing convex bodies from their projections,” Discr. Comput. Geom., 4, 205–237 (1989). · Zbl 0669.52001 · doi:10.1007/BF02187723
[157] Z. Lángi, ”On seven points on the boundary of a plane convex body in large relative distances,” Contrib. Alg. Geom., 45, 275–281 (2004). · Zbl 1052.52018
[158] M. Lassak, ”Approximation of convex bodies by parallelotopes,” Bull. Pol. Acad. Sci., Math., 39, 219–233 (1991). · Zbl 0756.52007
[159] M. Lassak, ”On five points in a plane convex body pairwise in at least unit relative distances,” Coll. Math. Soc. János Bolyai, Szegel, 69, 245–247 (1991). · Zbl 0822.52001
[160] M. Lassak, ”Approximation of convex bodies by triangles,” Proc. Amer. Math. Soc, 115, 207–210 (1992). · Zbl 0757.52008 · doi:10.1090/S0002-9939-1992-1057956-1
[161] M. Lassak, ”Approximation of convex bodies by rectangles,” Geom. Dedic., 47, 111–117 (1993). · Zbl 0779.52007 · doi:10.1007/BF01263495
[162] M. Lassak, ”Relationship between widths of a convex body and of an inscribed parallelotope,” Bull. Austr. Math. Soc., 63, 133–140 (2001). · Zbl 0989.52001 · doi:10.1017/S0004972700019195
[163] M. Lassak, ”Affine-regular hexagons of extreme areas inscribed in a centrally symmetric convex body,” Adv. Geom., 3, 45–51 (2003). · Zbl 1026.52004 · doi:10.1515/advg.2003.004
[164] M. Lassak, ”On relatively equilateral polygons inscribed in a convex body,” Publ. Math. Debrecen, 65, 133–148 (2004). · Zbl 1068.52003
[165] F. W. Levi, ”DeÜber zwei Sätze von Herrn Besicovitch,” Arch. Math., 3, 125–129 (1952). · Zbl 0048.16701 · doi:10.1007/BF01899353
[166] M. A. Lopez and S. Reisner, ”Efficient approximation of convex polygons,” Int. J. Comput. Geom. Appl., 10, 445–452 (2000). · Zbl 0973.65013 · doi:10.1142/S0218195900000267
[167] M. A. Lopez and S. Reisner, ”Linear time approximation of 3D convex polytopes,” Comput. Geom., 23, 291–301 (2002). · Zbl 1015.52013 · doi:10.1016/S0925-7721(02)00100-1
[168] L. Lovász and M. Simonovits, ”Random walks in a convex body and an improved volume algorithm,” Random Struct. Algorithms, 4, 359–412 (1993). · Zbl 0788.60087 · doi:10.1002/rsa.3240040402
[169] M. Ludwig, Asymptotische Approximation Konvexer Körper, Ph.D. Thesis, Techn. Univ. Vienna (1992).
[170] M. Ludwig, ”Asymptotic approximation of convex curves,” Arch. Math., 63, 377–384 (1994). · Zbl 0803.52004 · doi:10.1007/BF01189576
[171] M. Ludwig, ”Asymptotic approximation of convex curves: The Hausdorff metric case,” Arch. Math., 70, 331–336 (1998). · Zbl 0919.52006 · doi:10.1007/s000130050203
[172] M. Ludwig, ”A characterization of affine length and asymptotic approximation of convex discs,” Abh. Math. Semin. Univ. Hamb., 69, 75–88 (1999). · Zbl 0954.52002 · doi:10.1007/BF02940864
[173] M. Ludwig, ”Asymptotic approximation of smooth convex bodies by general polytopes,” Mathematica, 46, 103–125 (1999). · Zbl 0992.52002
[174] A. M. Macbeath, ”An extremal property of the hypersphere,” Proc. Cambrige Phil. Soc., 47, 245–247 (1951). · Zbl 0042.40801 · doi:10.1017/S0305004100026542
[175] A. M. Macbeath, ”Compactness theorem for affine equivalence classes of convex regions,” Can. J. Math., 3, 54–61 (1951). · Zbl 0042.16401 · doi:10.4153/CJM-1951-008-7
[176] V. V. Makeev, ”Inscribed and circumscribed polyhedra of convex bodies and related problems,” Mat. Zametki, 51, 67–71 (1992). · Zbl 0798.52007
[177] V. V. Makeev, ”Inscribed and circumscribed polyhedra of convex bodies,” Mat. Zametki, 55, 128–130 (1994). · Zbl 0830.52003
[178] V. V. Makeev, ”On approximation of planar sections of convex bodies,” Zap. Nauch. Semin. POMI, 246, 174–183 (1997). · Zbl 0910.52003
[179] V. V. Makeev, ”Three-dimensional inscribed and circumscribed polyhedra for convex compact sets,” Algebra Analiz, 12, 1–15 (2000).
[180] V. V. Makeev, ”Three-dimensional inscribed and circumscribed polyhedra for convex compact sets, 2,” Algebra Analiz, 13, 110–133 (2001).
[181] V. V. Makeev, ”On some combinatorial-geometric problems for vector bundles,” Algebra Analiz, 14, 169–191 (2002). · Zbl 1037.52010
[182] V. V. Makeev and A. S. Mukhin, ”On the existence of a common section for several convex compact sets that has prescribed properties,” Zap. Nauch. Semin. POMI, 261, 198–203 (1999).
[183] B. Massü, ”On the LLN for the number of vertices of a random convex hull,” Adv. Appl. Probab., 32, 675–681 (2000). · Zbl 0970.60010 · doi:10.1239/aap/1013540238
[184] D. E. McClure and R. A. Vitalie, ”Polygonal approximation of plane convex bodies,” J. Math. Anal. Appl., 51, 326–358 (1975). · Zbl 0315.52004 · doi:10.1016/0022-247X(75)90125-0
[185] J. S. Müller, ”Approximation of a ball by random polytopes,” J. Approx. Theory, 63, 198–209 (1990). · Zbl 0736.41027 · doi:10.1016/0021-9045(90)90103-W
[186] H. Niederreiter, ”On a measure of denseness for sequences,” in: Topics in Classical Number Theory, Colloq. Math. Soc. Janós Bolyai, 34 (1984), pp. 1163–1208.
[187] P. Novotný, ”Approximation of convex sets by simplexes,” Geom. Dedic., 50, 53–55 (1994). · Zbl 0807.52007 · doi:10.1007/BF01263651
[188] J. O’Rourke, ”Finding minimal enclosing boxes,” Int. J. Comput. Inform. Sci., 14, 183–199 (1985). · Zbl 0582.68067 · doi:10.1007/BF00991005
[189] J. O’Rourke, A. Aggarwal, S. Meddila, and M. Baldwin, ”An optimal algorithm for finding minimal enclosing triangle,” J. Algorithms, 7, 258–269 (1986). · Zbl 0606.68038 · doi:10.1016/0196-6774(86)90007-6
[190] A. Packer, ”Polynomial-time approximation of largest simplices in V-polytopes,” Discr. Appl. Math., 134, 213–237 (2004). · Zbl 1165.68524 · doi:10.1016/S0166-218X(03)00226-9
[191] O. Palmon, ”The only convex body with extremal distance from the ball is the simplex,” Isr. J. Math., 80, 337–349 (1992). · Zbl 0774.52003 · doi:10.1007/BF02808075
[192] A. Pelczynski and S. J. Szarek, ”On parallelepipeds of minimal volume containing a convex symmetric body in \(\mathbb{R}\)n,” Math. Proc. Camb. Phil. Soc., 109, 125–148 (1991). · Zbl 0718.52007 · doi:10.1017/S0305004100069619
[193] C. M. Petty, ”On the geometry of the Minkowski plane,” Riv. Mat. Univ. Parma, 6, 269–292 (1955). · Zbl 0067.40102
[194] V. A. Popov, ”Approximation of convex sets,” Izv. Mat. Inst. Bulg. Akad. Nauk, 11, 67–80 (1970). · Zbl 0223.52001
[195] K. Radziszewski, ”Sur un problème extremal relatif aux figures inscrites et circonscrite aux figures convexes,” Ann. Univ. M. Curie-Sclodowska, Sec. A, 6, 5–18 (1952). · Zbl 0053.12205
[196] S. Reisner, C. Schütt, E. Werner, ”Dropping a vertex or a facet from a convex polytope,” Forum Math., 13, 359–378 (2001). · Zbl 0980.52002 · doi:10.1515/form.2001.012
[197] M. Reitzner, ”Inequalities for convex hulls of random points,” Monatsh. Math., 131, 71–78 (2000). · Zbl 0982.52008 · doi:10.1007/s006050070025
[198] M. Reitzner, ”Stochastical approximation of smooth convex bodies,” in: Proc. Conf. ”Konvexgeometrie,” Oberwolfach, Apr. 22–28 2001, Tagungsber. Math. Forcshungsist. Oberwolfach, 18 (2001), p. 14.
[199] M. Reitzner, ”Random points on the boundary of smooth convex bodies,” Trans. Amer. Math. Soc., 354, 2243–2278 (2002). · Zbl 0993.60012 · doi:10.1090/S0002-9947-02-02962-8
[200] M. Reitzner, ”Random polytopes and the Efron-Stein jackknife inequality,” Ann. Probab., 31, 2136–2166 (2003). · Zbl 1058.60010 · doi:10.1214/aop/1068646381
[201] M. Reitzner, ”The combinatorial structure of random polytopes,” Adv. Math., 191, 178–208 (2005). · Zbl 1065.52004 · doi:10.1016/j.aim.2004.03.006
[202] A. Rényi, R. Sulanke, ”:Uber die konvexe Hülle von n zufällig gewahlten Punkten,” Z. Wahrscheinlichkeitsth, 2, 75–84 (1963). · Zbl 0118.13701 · doi:10.1007/BF00535300
[203] G. Rote, ”The convergence rate for the Sandwich algorithm for approximating convex figures in the plane,” in: Proc. 2nd Canad. Conf. on Comput. Geometry, Ottava (1990), pp. 287–290.
[204] G. Rote, ”The convergence rate of the Sandwich algorithm for approximating convex functions,” Computing, 48, 337–361 (1992). · Zbl 0787.65006 · doi:10.1007/BF02238642
[205] V. V. Ryabchenko, S. I. Lyashko, O. N. Grushets’kyj, and B. V. Rublyov, ”A new algorithm of linear complexity for finding triangles of maximal area inscribed in a convex polygon,” Visn. Kiev. Univ. Im. Tarasa Shevchenka, Ser. Fiz.-Mat. Nauky, 2, 210–214 (2003). · Zbl 1122.52303
[206] E. Sas, ”Ber eine Extremaleigenschalf der Ellipsen,” Compos. Math., 6, 468–470 (1939). · Zbl 0020.40201
[207] G. C. Shephard, ”Decomposable convex polyhedra,” Mathematica, 10, 89–95 (1963). · Zbl 0121.39002
[208] R. Schneider, ”Zwei Extremalaufgaben für konvexer Bereiche,” Acta Math. Acad. Sci. Hung., 22, 379–383 (1971). · Zbl 0233.52004 · doi:10.1007/BF01896433
[209] R. Schneider, ”Zur optimalen Approximation konvexer Hyperflachen durch Polyeder,” Math. Ann., 256, 289–301 (1981). · Zbl 0458.52003 · doi:10.1007/BF01679698
[210] R. Schneider, ”Affine-invariant approximation by convex polytopes,” Studia Sci. Math. Hung., 21, 401–408 (1986). · Zbl 0552.52001
[211] R. Schneider, ”Polyhedral approximation of smooth convex bodies,” J. Math. Anal. Appl., 128, 470–474 (1987). · Zbl 0629.52006 · doi:10.1016/0022-247X(87)90197-1
[212] R. Schneider, ”Random approximation of convex sets,” J. Microscopy, 151, 211–227 (1988). · Zbl 1256.52004
[213] R. Schneider and J. A. Wieacker, ”Approximation of convex bodies by polytopes,” Bull. London Math. Soc., 13, 149–156 (1981). · Zbl 0445.52003 · doi:10.1112/blms/13.2.149
[214] C. Schütt, ”The convex floating body and polyhedral approximation,” Isr. J. Math., 73, 65–77 (1991). · Zbl 0745.52006 · doi:10.1007/BF02773425
[215] O. Schwarzkopf, U. Futchs, G. Rote, and E. Welzl, ”Approximation of convex figures by pairs of rectangles,” Comput. Geom., 10, 77–87 (1998). · Zbl 0896.68144 · doi:10.1016/S0925-7721(96)00019-3
[216] V. N. Shevchenko, Qualitative Problems of Integer-Valued Linear Programming [in Russian], Fizmatlit, Nauka (1995). · Zbl 0947.90076
[217] Ya. G. Sinai, ”Probabilistic approach to the analysis of statistics of convex broken lines,” Funkts. Anal. Prilozh., 28, 16–25 (1994).
[218] G. Sonnevand, ”An optimal sequential algorithm for the uniform approximation of convex functions on [0,1]2,” Appl. Math. Optim., 10, 127–142 (1983). · Zbl 0527.65010 · doi:10.1007/BF01448382
[219] W. Süss, ”Über Parallelogramme und Rechtecke, die sich ebenen Eibereichen einschreben lassen,” Rend. Math. Appl., 14, 338–341 (1955). · Zbl 0065.15501
[220] S. Tabachnikov, ”On the dual billiard problem,” Adv. Math., 115, 221–249 (1995). · Zbl 0846.58038 · doi:10.1006/aima.1995.1055
[221] P. Valtr, ”Probability that n random points are in convex position,” Discr. Comput. Geom. 13, 637–643 (1995). · Zbl 0820.60007 · doi:10.1007/BF02574070
[222] P. Valtr, ”The probability that n random points in a triangle are in convex position,” Combinatorica, 16, 567–573 (1996). · Zbl 0881.60010 · doi:10.1007/BF01271274
[223] A. M. Vershik, ”Limit forms of convex integer-valued polygons and related topics,” Funkts. Anal. Prilozh., 28, 16–25 (1994). · Zbl 0848.52004
[224] A. M. Vershik, ”Limit forms of typical geometric configurations and their applications,” Zap. Nauch. Semin. POMI, 280, 73–100 (2001). · Zbl 1146.51303
[225] A. M. Vershik and P. V. Sporyshev, ”Asymptotic behavior of the number of faces of random polyhedra and the neighborliness problem,” Sel. Math. Sov., 11, 181–201 (1992). · Zbl 0791.52011
[226] A. M. Vershik and O. Zeitouni, ”Large deviations in the geometry of complex lattice polygons,” Isr. J. Math., 109, 13–27 (1999). · Zbl 0945.60022 · doi:10.1007/BF02775023
[227] J. A. Wieaker, ”Eintre Probleme der polyedrishen Approximation,” Diplomarbeit, Univ. Freiburg (1978).
[228] N. V. Zhivkov, ”Plane polygonal approximation of bounded convex sets,” Dokl. Bulg. Akad. Nauk, 35, 1631–1634 (1982). · Zbl 0518.41017
[229] Y. Zhou and S. Suri, ”Algorithms for minimum volume enclosing a simplex in three dimensions,” SIAM J. Comput., 31, No. 5, 1339–1357 (2002). · Zbl 1041.68052 · doi:10.1137/S0097539799363992
[230] G. M. Ziegler, ”Lectures on 0/1 polytopes,” in: Polytopes: Combinatorics and Computation, DMV-Seminars, Birkhäuser (2000), pp. 1–44. · Zbl 0966.52012
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