On the symmetry function of a convex set

Abstract

We attempt a broad exploration of properties and connections between the symmetry function of a convex set S ${S \subset\mathbb{R}^n}$ and other arenas of convexity including convex functions, convex geometry, probability theory on convex sets, and computational complexity. Given a point ${x \in S}$, let sym(x,S) denote the symmetry value of x in S:

${{\bf sym}(x,S):= max\{\alpha \ge 0 : x+\alpha(x-y) \in S {\rm for every} y \in S\}}$, which essentially measures how symmetric S is about the point x, and define ${\bf sym}({\it S}):= \max_{x\in S} \, {\bf sym}({\it x,S})$ x ^* is called a symmetry point of S if x ^* achieves the above maximum. The set S is a symmetric set if sym (S)=1. There are many important properties of symmetric convex sets; herein we explore how these properties extend as a function of sym (S) and/or sym (x,S). By accounting for the role of the symmetry function, we reduce the dependence of many mathematical results on the strong assumption that S is symmetric, and we are able to capture and otherwise quantify many of the ways that the symmetry function influences properties of convex sets and functions. The results in this paper include functional properties of sym (x,S), relations with several convex geometry quantities such as volume, distance, and cross-ratio distance, as well as set approximation results, including a refinement of the Löwner-John rounding theorems, and applications of symmetry to probability theory on convex sets. We provide a characterization of symmetry points x ^* for general convex sets. Finally, in the polyhedral case, we show how to efficiently compute sym(S) and a symmetry point x ^* using linear programming. The paper also contains discussions of open questions as well as unproved conjectures regarding the symmetry function and its connection to other areas of convexity theory.