{"ID":5937941,"CreatedAt":"2026-07-07T03:14:33.014478982Z","UpdatedAt":"2026-07-07T10:45:56.18268788Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.03815","arxiv_id":"2607.03815","title":"A simplex-based measure of symmetry","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.","short_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 resu...","url_abs":"https://arxiv.org/abs/2607.03815","url_pdf":"https://arxiv.org/pdf/2607.03815v1","authors":"[\"Egor Bakaev\",\"Amir Yehudayoff\"]","published":"2026-07-04T10:50:29Z","proceeding":"math.MG","tasks":"[\"math.MG\",\"cs.LG\"]","methods":"[]","has_code":false}