Shadow systems, decomposability and isotropic constants

TL;DR

This paper investigates conditions under which local maximizers of the isotropic constant have limited decomposability, revealing that their polar bodies can only be decomposed into a bounded number of Minkowski summands, with implications for convex geometry.

Contribution

It establishes a bound on the decomposability of polar bodies of local maximizers of the isotropic constant and extends existing results to shadow systems and bodies with symmetries.

Findings

Polar bodies of local maximizers have decomposability dimension at most (n^2+3n)/2

Extension of RS-decomposability results to broader shadow systems

Connections between polytopal cases and affine rigidity theory

Abstract

We study necessary conditions for local maximizers of the isotropic constant that are related to notions of decomposability. Our main result asserts that the polar body of a local maximizer of the isotropic constant can only have few Minkowski summands; more precisely, its dimension of decomposability is at most $\frac12(n^2+3n)$. Using a similar proof strategy, a result by Campi, Colesanti and Gronchi concerning RS-decomposability is extended to a larger class of shadow systems. We discuss the polytopal case, which turns out to have connections to (affine) rigidity theory, and investigate how the bound on the maximal number of irredundant summands can be improved if we restrict our attention to convex bodies with certain symmetries.