用单纯形定义对称性度量，揭示凸体与单纯形的逼近关系。
A simplex-based measure of symmetry
- 基于单纯形构造对称性度量，具仿射不变性。
- 若对称性接近n，凸体与单纯形的Banach-Mazur距离在1/(1-ε)内。
- 首次刻画单纯形为唯一满足外可加性的凸体，关联神经网络深度复杂度。
对于紧致凸集 $L,K \subset \mathbb{R}^n$，记 $λ_K(L)$ 为包含 $L$ 所需最小同胚 $K$ 的尺度。本文定义以 $n$-单纯形 $Δ= Δ^n \subset \mathbb{R}^n$ 为基础的对称性度量 $ρ_Δ(L):=\frac{λ_{-Δ}(L)}{λ_Δ(L)}$。研究发现：(1) 经典Minkowski对称性度量 $m^*(L)$ 可视为 $ρ_Δ(L)$ 的仿射不变版本；(2) 改进稳定性分析：若 $m^*(L)\ge n-\varepsilon$，则 $L$ 与 $Δ$ 在Banach--Mazur距离下 $\tfrac{1}{1-\varepsilon}$-接近；(3) 首次证明单纯形是唯一使函数 $L \mapsto λ_K(L)$ 满足外可加性的凸体；(4) 受ReLU神经网络表达性启发，研究 $\mathbb{R}^n$ 中多面体的深度复杂度，证明每个深度复杂度为 $d$ 的多面体 $P$ 满足 $ρ_Δ(P) \leq 2^d -1$，即低深度多面体无法逼近单纯形。
原文摘要 · Abstract (English)
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.
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