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Inequalities for polars of mixed projection bodies. (English) Zbl 1070.52006

The following are the main results of the paper. Let \(K_1,\dots,K_{n-1},K,L\) be convex bodies in \({\mathbb R}^n\), let \(r \in \{2,\dots,n-1\}\) and \(\lambda,\mu\geq 0\). \(V\) denotes volume. Then \[ V(\Pi^*(K_1,\dots,K_{n-1}))^r\leq\prod_{j=1}^r V(\Pi^*(\underbrace{K_j,\dots,K_j}_r,K_{r+1},\dots,K_{n-1}), \] with equality if \(K_1,\dots,K_{n-1}\) are homothetic. If \(C=(K_2,\dots,K_{n-1})\), then \[ 4V(\Pi^*(\lambda K+\mu L,C))^{1/n} \leq \lambda V(\Pi^*(K,C))^{1/n}+\mu V(\Pi^*(L,C))^{1/n}, \] with equality if and (for \(\lambda,\mu>0\)) only if \(\Pi(K,C)=\Pi(L,C)\). Here \(\Pi(K_1,\dots,K_{n-1})\) denotes the mixed projection body of \(K_1,\dots,K_{n-1}\), as introduced by E. Lutwak [Trans. Am. Math. Soc. 339, 901–916 (1993; Zbl 0784.52009)], and \(\Pi^*\) is the polar body of \(\Pi\). The proof combines the Aleksandrov-Fenchel inequalities with the integral inequalities of Hölder and Minkowski. A further result extends W. Firey’s [Math. Scand. 8, 168–170 (1960; Zbl 0104.17003)] Pythagorean inequalities for projection volumes.

MSC:

52A40 Inequalities and extremum problems involving convexity in convex geometry
52A39 Mixed volumes and related topics in convex geometry
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References:

[1] Lutwak, E., Inequalities for mixed projection bodies, Tans. Amer. Math. Soc., 1993, 339: 901-916. · Zbl 0784.52009
[2] Bolker, E. D., A class of convex bodies, Trans. Amer. Math. Soc., 1969, 145: 323-345. · Zbl 0194.23102
[3] Brannen, N. S., Volumes of projection bodies, Mathematika, 1996, 43: 255-264. · Zbl 0872.52004
[4] Schneider, R., Convex Bodies: The Brunn-Minkowski Theory, Cambridge: Cambridge University Press, 1993. · Zbl 0798.52001
[5] Lutwak, E., Centroid bodies and dual mixed volumes, Proc. London Math. Soc., 1990, 60: 365-391. · Zbl 0703.52005
[6] Bourgain, J., Lindenstrauss, J., Projection bodies, Israel Seminar (G. A. F. A.)1986-1987, Lecture Notes in Math, Vol.1317, Berlin, New york: Springer-Verlag, 1988, 250-269. · Zbl 0645.52002
[7] Ball, K., Shadows of convex bodies, Tans. Amer. Math. Soc., 1991, 327: 891-901. · Zbl 0746.52007
[8] Chakerian, G. D., Lutwak, E., Bodies with similar projections, Tans. Amer. Math. Soc., 1997, 349: 1811-1820. · Zbl 0873.52010
[9] Witsenhausen, H. S., A support characterization of the zonotopes, Mathematika, 1978, 25: 13- 16. · Zbl 0368.52008
[10] Gordon, Y., Meyer, M., Reisner, S., Zonoids with minimal volume produt-a new proof, Pro. Amer. Math. Soc., 1988,104: 273-276. · Zbl 0663.52003
[11] Lutwak, E., Intersection bodies and dual mixed volumes, Adv. Math., 1988, 71: 232-261. · Zbl 0657.52002
[12] Reisner, S., Zonoids with minimal volume-produt, Math. Zeitschr, 1986, 192: 339-346. · Zbl 0578.52005
[13] Stanley, R. P., Two combinatorial applications of the Aleksandrov-Fenchel inequalities, J. Combin. Theory, Ser. A, 1981, 31: 56-65. · Zbl 0484.05012
[14] Betke, U., McMullen, P., Estimating the sizes of convex bodies from projection. J. London Math. Soc., 1983, 27: 525-538. · Zbl 0487.52005
[15] Schneider, R., Random ploytopes generated by anisotropic hyperplanes, Bull. Lodon Math. Soc., 1982, 14: 549-553. · Zbl 0476.60010
[16] Vitale, R. A., Expected absolute random determinants and zonoids, Ann. Appl. Probab., 1991,1: 293-300. · Zbl 0739.15015
[17] Alexander, R., Zonoid theory and Hilbert’s fouth problem, Geom. Dedicata., 1988, 28: 199-211. · Zbl 0659.51022
[18] Goodey, P. R., Weil, W., Zonoids and generalizations, in Handbook of Convex Geometry (ed. Gruber, P. M., Wills, J. M.), North-Holland, Amsterdam, 1993, 326: 1297. · Zbl 0791.52006
[19] Martini, H., Zur Bestimmung Konvexer Polytope durch the Inhalte ihrer Projection, Beiträge Zur Algebra und Geometrie, 1984,18: 75-85. · Zbl 0544.52005
[20] Schneider, R., Weil, W., Zonoids and related topics, Convexity and its Applications, Basel: Birkhäuser, 1983, 296-317.
[21] Bonnesen, T., Fenchel, W., Theorie der Konvexen Körper, Berlin: Springer, 1934. · Zbl 0008.07708
[22] Chakerian, G. D., Set of constant relative width and constant relative brightness, Trans. Amer. Math. Soc., 1967, 129: 26-37. · Zbl 0161.41606
[23] Lutwak, E., On quermassintegrals of mixed projection bodies, Geom. Dedicata, 1990, 33: 51-58. · Zbl 0692.52006
[24] Lutwak, E., Volume of mixed bodies, Trans. Amer. Math. Soc., 1986, 294: 487-500. · Zbl 0591.52016
[25] Lutwak, E., Mixed projection inequalities, Trans. Amer. Math. Soc., 1985, 287: 91-106. · Zbl 0555.52010
[26] Firey, W. J., Pythagorean inequalities for convex bodies, Math. Scand., 1960, 8: 168-170. · Zbl 0104.17003
[27] Leng Gangsong, Zhang Liansheng, Extreme Properties of quermassintegrals of convex bodies, Science in China, Ser. A, 2001, 44: 837-845. · Zbl 1006.52003
[28] Blaschke, W., Vorlesungen über integralgeometrie, I, II, Teubner, Leipzig, 1936, 1937; reprint, New York: Chelsea, 1949. · JFM 63.0675.04
[29] Lutwak, E., Width-integrals of convex bodies, Proc. Amer. Math. Soc., 1975, 53: 435-439. · Zbl 0276.52006
[30] Ren, D. L., An Introduction to Integral Geometry (in Chinese), Shanghai: Science and Technology Press, 1988.
[31] Hardy, G. H., Littlewood, J. E., Pólya, G., Inequalities, Cambridge: Cambridge University Press, 1934.
[32] Rogers, C. A., Setions and projection of convex bodies, Portugal Math., 1965, 24: 99-103. · Zbl 0137.15401
[33] Richard, J., Gardner, Geometric Tomography, Cambridge: Cambridge University Press, 1995.
[34] Barth, Aextremal property of the bmean width of the simplex, Math. Ann., 1998, 310: 685-693. · Zbl 0901.52013
[35] Kawashima, Polytopes which are orthogonal projections of regular simplexes, Geom. Dedicata, 1991, 38: 73-85. · Zbl 0722.52007
[36] Yang, L., Zhang, J. Z., Pseudo-symmetric point set geometric inequalities, Acta Math. Sinica, 1986, 6: 802-806. · Zbl 0651.52012
[37] Zhang, J. Z., Yang, L., The equelspaced embedding of finite point set in pseudo-Eucldeau space, Acta Math. Sinica, 1981, 24: 481-487. · Zbl 0522.51016
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