An estimate of the volume of a compact set in terms of its integral of mean curvature (Q1122144)

A geometric inequality for a compact set in \({\mathbb{R}}^ 3\) is obtained and some properties of the set are studied. We prove that if W is M(\(\infty)\)-compact set, then (volume of \(W)\leq (1/48\pi^ 2)\cdot (integral\) of mean curvature of \(W)^ 3\) and if K is compact convex subset of M(\(\infty)\)-compact set, then (integral of mean curvature of K)\(\leq (integral\) of mean curvature of W). The method of outer parallel bodies is used in the proof.