The isotropy constant and boundary properties of convex bodies
HTML articles powered by AMS MathViewer
- by Mathieu Meyer and Shlomo Reisner PDF
- Proc. Amer. Math. Soc. 144 (2016), 3935-3947 Request permission
Abstract:
Let $\mathcal {K}^n$ be the set of all convex bodies in $\mathbb {R}^n$ endowed with the Hausdorff distance. We prove that if $K\in \mathcal {K}^n$ has positive generalized Gauss curvature at some point of its boundary, then $K$ is not a local maximizer for the isotropy constant $L_K$.References
- Gergely Ambrus and Károly J. Böröczky, Stability results for the volume of random simplices, Amer. J. Math. 136 (2014), no. 4, 833–857. MR 3245182, DOI 10.1353/ajm.2014.0030
- J. Bourgain, On the distribution of polynomials on high-dimensional convex sets, Geometric aspects of functional analysis (1989–90), Lecture Notes in Math., vol. 1469, Springer, Berlin, 1991, pp. 127–137. MR 1122617, DOI 10.1007/BFb0089219
- Silouanos Brazitikos, Apostolos Giannopoulos, Petros Valettas, and Beatrice-Helen Vritsiou, Geometry of isotropic convex bodies, Mathematical Surveys and Monographs, vol. 196, American Mathematical Society, Providence, RI, 2014. MR 3185453, DOI 10.1090/surv/196
- Stefano Campi, Andrea Colesanti, and Paolo Gronchi, A note on Sylvester’s problem for random polytopes in a convex body, Rend. Istit. Mat. Univ. Trieste 31 (1999), no. 1-2, 79–94. MR 1763244
- Yehoram Gordon and Mathieu Meyer, On the minima of the functional Mahler product, Houston J. Math. 40 (2014), no. 2, 385–393. MR 3248645
- B. Klartag, On convex perturbations with a bounded isotropic constant, Geom. Funct. Anal. 16 (2006), no. 6, 1274–1290. MR 2276540, DOI 10.1007/s00039-006-0588-1
- V. D. Milman and A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed $n$-dimensional space, Geometric aspects of functional analysis (1987–88), Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 64–104. MR 1008717, DOI 10.1007/BFb0090049
- Luis Rademacher, A simplicial polytope that maximizes the isotropic constant must be a simplex, Mathematika 62 (2016), no. 1, 307–320. MR 3430385, DOI 10.1112/S0025579315000133
- Shlomo Reisner, Carsten Schütt, and Elisabeth M. Werner, Mahler’s conjecture and curvature, Int. Math. Res. Not. IMRN 1 (2012), 1–16. MR 2874925, DOI 10.1093/imrn/rnr003
- Carsten Schütt and Elisabeth Werner, The convex floating body, Math. Scand. 66 (1990), no. 2, 275–290. MR 1075144, DOI 10.7146/math.scand.a-12311
Additional Information
- Mathieu Meyer
- Affiliation: Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées UMR 8050, UPEMLV, UPEC, CNRS F-77454, Marne-la-Vallée, France
- MR Author ID: 197612
- Email: mathieu.meyer@u-pem.fr
- Shlomo Reisner
- Affiliation: Department of Mathematics, University of Haifa, Haifa, 31905, Israel
- MR Author ID: 146685
- Email: reisner@math.haifa.ac.il
- Received by editor(s): November 12, 2015
- Published electronically: April 25, 2016
- Communicated by: Thomas Schlumprecht
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 3935-3947
- MSC (2010): Primary 46B20, 52A20, 53A05
- DOI: https://doi.org/10.1090/proc/13143
- MathSciNet review: 3513550