Inequalities for the Surface Area of Projections of Convex Bodies

Giannopoulos, Apostolos ; Koldobsky, Alexander ; Valettas, Petros

Canadian journal of mathematics, Tome 70 (2018) no. 4, pp. 804-823

Résumé

We provide general inequalities that compare the surface area $S(K)$ of a convex body $K$ in ${{\mathbb{R}}^{n}}$ to the minimal, average, or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for all the quermassintegrals of $K$ . We examine separately the dependence of the constants on the dimension in the case where $K$ is in some of the classical positions or $K$ is a projection body. Our results are in the spirit of the hyperplane problem, with sections replaced by projections and volume by surface area.

Détail

DOI : 10.4153/CJM-2016-051-x

Mots-clés : 52A20, 46B05, surface area, convex body, projection

Bibliographie

[1] [1] Artstein-Avidan, S., Giannopoulos, A., and Milman, V. D., Asymptotic geometric analysis. Vol. I. Mathematical Surveys and Monographs, 202, American Mathematical Society, Providence, RI, 2015. http://dx.doi.Org/10.1090/surv/202 Google Scholar

[2] [2] Ball, K. M., Volume ratios and a reverse isoperimetric inequality. J. London Math. Soc. (2) 44(1991), no. 2, 351–359. http://dx.doi.Org/10.1112/jlms/s2-44.2.351 Google Scholar

[3] [3] Brazitikos, S., Giannopoulos, A., Valettas, P., and Vritsiou, B.-H., Geometry ofisotropic convex bodies. Mathematical Surveys and Monographs, 196, American Mathematical Society, Providence, RI, 2014. http://dx.doi.Org/10.1090/surv/196 Google Scholar

[4] [4] Burago, Y. D. and Zalgaller, V. A., Geometric inequalities. Springer Series in Soviet Mathematics, Springer-Verlag, Berlin-New York, 1988. Google Scholar | DOI

[5] [5] Fradelizi, M., Giannopoulos, A., and Meyer, M., Some inequalities about mixed volumes. Israel J. Math. 135(2003), 157–179. Google Scholar | DOI

[6] [6] Gardner, R. J., Geometric tomography. Second Ed., Encyclopedia of Mathematics and its Applications, 58, Cambridge University Press, Cambridge, 2006. http://dx.doi.Org/10.1017/CBO9781107341029 Google Scholar

[7] [7] Giannopoulos, A., Hartzoulaki, M., and Paouris, G., On a local version of the Aleksandrov-Fenchel inequality for the quermassintegrals of a convex body. Proc. Amer. Math. Soc. 130(2002), no. 8, 2403–2412. Google Scholar | DOI

[8] [8] Giannopoulos, A. and Milman, V. D., Extremal problems and isotropic positions of convex bodies. Israel J. Math. 117(2000), 29–60. Google Scholar | DOI

[9] [9] Giannopoulos, A. and Papadimitrakis, M., Isotropic surface area measures. Mathematika 46(1999), 1–13. Google Scholar | DOI

[10] [10] Koldobsky, A., Stability and separation in volume comparison problems. Math. Model. Nat. Phenom. 8(2013), 156–169. Google Scholar | DOI

[11] [11] Koldobsky, A., Stability inequalities for projections of convex bodies. Discrete Comput. Geom. 57(2017), 152–163. Google Scholar | DOI

[12] [12] Markessinis, E., Paouris, G., and Saroglou, Ch., Comparing the M-position with some classical positions of convex bodies. Math. Proc. Cambridge Philos. Soc. 152(2012), 131–152. http://dx.doi.Org/10.1017/S0305004111000703 Google Scholar

[13] [13] Petty, C. M., Surface area of a convex body under affme transformations. Proc. Amer. Math. Soc. 12(1961), 824–828. Google Scholar | DOI

[14] [14] Schneider, R., Convex bodies: The Brunn-Minkowski theory. Second expanded ed., Encyclopedia of Mathematics and its Applications, 151, Cambridge University Press, Cambridge, 2014. Google Scholar

[15] [15] Schneider, R. and Weil, W., Zonoids and selected topics. In: Convexity and its Applications, Birkhauser, Boston, Mass., 1983. Google Scholar