A simplex-based measure of symmetry

2026-07-04

AI Summary

This work proposes a novel symmetry measure ρ_Δ(L) for convex bodies based on n-dimensional simplices, defined as the ratio of the smallest homothetic simplex containing L. This metric refines the stability analysis of classical Minkowski symmetry and establishes, for the first time, that the simplex is the unique convex body satisfying the "outer additivity" property. By integrating tools from convex geometry, affine invariants, and the Banach–Mazur distance, the study connects ρ_Δ to the expressive power of ReLU neural networks: if ρ_Δ(L) ≥ n−ε, then L is close to a simplex in the Banach–Mazur sense; conversely, any polytope P representable by a ReLU network of depth d satisfies ρ_Δ(P) ≤ 2^d −1, implying that simplices cannot be efficiently approximated by low-depth polytopes.

Abstract

For compact convex sets $L,K \subset \mathbb{R}^n$, denote by $λ_K(L)$ the smallest size of a homothet of $K$ that contains $L$. We define a measure of symmetry based on the $n$-simplex $Δ= Δ^n \subset \mathbb{R}^n$ 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 $m^*(L)$ can be defined as an affine-invariant version of $ρ_Δ(L)$. (2) We improve the stability analysis for the Minkowski measure of symmetry; if $m^*(L)\ge n-\varepsilon$ then $L$ is $\tfrac{1}{1-\varepsilon}$-close to $Δ$ in the Banach--Mazur distance. (3) We obtain a novel characterization of simplices as the only convex bodies $K$ for which the function $L \mapsto λ_K(L)$ 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 $ρ_Δ(P) \leq 2^d -1$ for every polytope $P$ of depth complexity $d$. In other words, simplices cannot be approximated by low-depth polytopes.

Problem

• symmetry measure
• convex bodies
• simplex
• Minkowski sum
• depth complexity

Innovation

• simplex-based symmetry measure
• outer additivity
• Minkowski measure stability
• depth complexity of polytopes
• Banach–Mazur distance