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A SIMPLICIAL POLYTOPE THAT MAXIMIZES THE ISOTROPIC CONSTANT MUST BE A SIMPLEX

Part of: Metric geometry General convexity Optimality conditions

Published online by Cambridge University Press:  22 May 2015

Luis Rademacher
Luis Rademacher*
Affiliation:
Computer Science and Engineering, The Ohio State University, U.S.A. email lrademac@cse.ohio-state.edu

Abstract

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The isotropic constant $L_{K}$ is an affine-invariant measure of the spread of a convex body $K$. For a $d$-dimensional convex body $K$, $L_{K}$ can be defined by $L_{K}^{2d}=\det (A(K))/(\text{vol}(K))^{2}$, where $A(K)$ is the covariance matrix of the uniform distribution on $K$. It is an open problem to find a tight asymptotic upper bound of the isotropic constant as a function of the dimension. It has been conjectured that there is a universal constant upper bound. The conjecture is known to be true for several families of bodies, in particular, highly symmetric bodies such as bodies having an unconditional basis. It is also known that maximizers cannot be smooth. In this work we study bodies that are neither smooth nor highly symmetric by showing progress towards reducing to a highly symmetric case among non-smooth bodies. More precisely, we study the set of maximizers among simplicial polytopes and we show that if a simplicial polytope $K$ is a maximizer of the isotropic constant among $d$-dimensional convex bodies, then when $K$ is put in isotropic position it is symmetric around any hyperplane spanned by a $(d-2)$-dimensional face and the origin. By a result of Campi, Colesanti and Gronchi, this implies that a simplicial polytope that maximizes the isotropic constant must be a simplex.

MSC classification

Primary: 52A23: Asymptotic theory of convex bodies
Secondary: 52A40: Inequalities and extremum problems 49K30: Optimal solutions belonging to restricted classes 51F15: Reflection groups, reflection geometries

Information

Type
Research Article
Information
Mathematika , Volume 62 , Issue 1 , 2016 , pp. 307 - 320
Copyright
Copyright © University College London 2015 

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