Isotropic constants and regular polytopes

TL;DR

This paper investigates the structure of polytopes that maximize the isotropic constant, revealing that certain local maximizers must be simplices, cross-polytopes, or cubes, depending on their properties.

Contribution

It establishes new structural characterizations of polytopal maximizers of the isotropic constant, extending previous results and analyzing specific classes like zonotopes.

Findings

A polytopal local maximizer with a simplicial vertex must be a simplex.

A centrally symmetric local maximizer with a simplicial vertex must be a cross-polytope.

A zonotope with a cubical zone that maximizes the isotropic constant must be a cube.

Abstract

We discuss first-order optimality conditions for the isotropic constant and combine them with RS-movements to obtain structural information about polytopal maximizers. Strengthening a result by Rademacher, it is shown that a polytopal local maximizer with a simplicial vertex must be a simplex. A similar statement is shown for a centrally symmetric local maximizer with a simplicial vertex: it has to be a cross-polytope. Moreover, we show that a zonotope that maximizes the isotropic constant and that has a cubical zone must be a cube. Finally, we consider the class of zonotopes with at most n+1 generators and determine the extremals in this class.