A simplex-based measure of symmetry

For compact convex sets L,K subset ℝn, denote by λK(L) the smallest size of a homothet of K that contains L. We define a measure of symmetry based on the n-simplex Delta = Deltan subset ℝn as the ratio [ ρ_Delta(L):=fracλ-Delta(L)λDelta(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 ρ_Delta(L). (2) We improve the stability analysis for the Minkowski measure of symmetry; if m^*(L)≥ n-varepsilon then L is tfrac11-varepsilon-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 λK(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 ℝn under the two operations: Minkowski sum and convex hull of a union. We prove the sharp bound ρ_Delta(P) ≤ 2d -1 for every polytope P of depth complexity d. In other words, simplices cannot be approximated by low-depth polytopes.