A simplex-based measure of symmetry

arXiv:2607.03815v1 Announce Type: cross Abstract: For compact convex sets L,K subset R^n, denote by lambda_K(L) the smallest size of a homothet of K that contains L. We define a measure of symmetry based on the n-simplex Delta = Delta^n subset R^n as the ratio [ rho_Delta(L):=frac{lambda_{-Delta}(L)}{lambda_{Delta}(L)}. ] We study this measure and deduce the following results: (1) The classical Minkowski measure of symmetry m^(L) can be defined as an affine-invariant version of rho_Delta(L). (2) We improve the stability analysis for the Minkowski measure of symmetry; if m^(L)ge n-arepsilon then L is frac{1}{1-arepsilon}-close to Delta 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 lambda_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 R^n under the two operations: Minkowski sum and convex hull of a union. We prove the sharp bound rho_Delta(P) leq 2^d -1 for every polytope P of depth complexity d. In other words, simplices cannot be approximated by low-depth polytopes.

Source: arXiv cs.LG | 2026-07-07