Isoperimetric inequality

The isoperimetric inequality is a fundamental principle of geometry stating that among all regions with a given boundary measure, the ball encloses the greatest volume — or equivalently, among all regions with a given volume, the ball has the smallest surface area. This principle appears across mathematics with remarkable universality: in Euclidean space it governs the relationship between perimeter and area; on curved manifolds it encodes Ricci curvature information through the Levy-Gromov comparison; in probability theory it underpins concentration of measure via log-Sobolev inequalities; and in graph theory it controls expansion and mixing properties of Markov chains. The inequality is not merely a geometric curiosity but a probe into the deep connection between local curvature constraints and global extremal structure — a connection that Mikhail Gromov exploited to develop his theory of metric measure spaces and that continues to bridge metric geometry, functional analysis, and probability theory.