Stability Properties of Geometric Inequalities

Abstract

Geometric inequalities, like the isoperimetric inequality, have been a subject of intensive research for a long time. More recently, some properties of these inequalities, which may be called stability properties, have been investigated. Roughly speaking, these investigations concern the geometric implications if the inequalities are in a certain sense close to equalities. The present article is primarily an exposition of various stability results concerning inequalities for plane convex sets, including a discussion of the concepts and methods which are of importance in this area. Moreover, several new results and proofs are presented. 1. Introduction. Let RI denote n-dimensional Eucidean space. A bounded con- vex subset of R will be called an n-dimensional convex body if it is closed and has interior points. We let qpn denote the class of all n-dimensional convex bodies. Two-dimensional convex bodies will be called convex domains. We consider primarily inequalities of the form