Isoperimetric Problems for Convex Bodies and a Localization Lemama

1 LL aszll o Lovv asz 2 and Mikll os Simonovits 3 Abstract. We study the smallest number (K) such that a given convex body K in IR n can be cut into two parts K 1 and K 2 by a surface with an (n ? 1)-dimensional measure (K)vol(K 1) vol(K 2)=vol(K). Let M 1 (K) be the average distance of a point of K from its center of gravity. We prove for the \isoperimetric coeecient" that and give other upper and lower bounds. We conjecture that our upper bound is best possible up to a constant. Our main tool is a general \Localization Lemma" that reduces integral inequalities over the n-dimensional space to integral inequalities in a single variable. This lemma was rst proved by two of the authors in an earlier paper, but here we give various extensions and variants that make its application smoother. We illustrate the usefulness of the lemma by showing how a number of well-known results can be proved using it. 1. Isoperimetry in a convex body The classical isoperimetric problem is to nd a surface with minimal measure which encloses a set of (at least) a given volume. We consider a \relativized" version of this problem, where we are given a convex body K, and want to nd a surface which divides K into two parts, and whose measure is minimum relative to the volumes of the two parts. To be more precise, (a) The (n ? 1){dimensional Minkowski measure of a set A IR n is deened as the limit (if it exists) of the volume of the "=2{neighbourhood of A divided by ", when " ! 0. (Volume means Lebesque measure.) Deene the isoperimetric coeecient of a convex body K IR n as the largest number = (K) such that for every measurable subset S K for which @S \ K (@S is the boundary of S) has an (n ? 1)-dimensional Minkowski measure, vol n?1 (@S \ K) vol(S) vol(K n S)