A simplex-based measure of symmetry

Abstract

For compact convex sets $L,K\subset {\mathbb{R}}^n$, denote by $\hat{I}{»}_K(L)$ the smallest size of a homothet of $K$ that contains $L$. We define a measure of symmetry based on the $n$-simplex $\hat{I}”=\hat{I}{”}^n\subset {\mathbb{R}}^n$ as the ratio \[ ρ_Δ(L):=\frac{λ_{-Δ}(L)}{λ_Δ(L)}. \] We study this measure and deduce the following results: (1) The classical Minkowski measure of symmetry $m^{\ast }(L)$ can be defined as an affine-invariant version of $\ddot{I}{}_{\hat{I}}”(L)$. (2) We improve the stability analysis for the Minkowski measure of symmetry; if $m^{\ast }(L)\ge n-\epsilon$ then $L$ is $\frac{1}{1-\epsilon }$-close to $\hat{I}”$ in the Banach--Mazur distance. (3) We obtain a novel characterization of simplices as the only convex bodies $K$ for which the function $L\mapsto \hat{I}{»}_K(L)$ is additive (a property we term ``outer additivity''). (4) Motivated by the expressivity of ReLU neural networks, we study the depth complexity of polytopes in ${\mathbb{R}}^n$ under the two operations: Minkowski sum and convex hull of a union. We prove the sharp bound $\ddot{I}{}_{\hat{I}}”(P)\le 2^d-1$ for every polytope $P$ of depth complexity $d$. In other words, simplices cannot be approximated by low-depth polytopes.