Isoperimetric problems for convex bodies and a localization lemma

Abstract

We study the smallest number ψ(K) such that a given convex bodyK in ℝ n can be cut into two partsK 1 andK 2 by a surface with an (n−1)-dimensional measure ψ(K) vol(K 1)·vol(K 2)/vol(K). LetM 1(K) be the average distance of a point ofK from its center of gravity. We prove for the “isoperimetric coefficient” that % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaqegWuDJLgzHbYqV52CVXwzaGGbciaa-H8acqGGOaak% cqWGlbWscqGGPaqkcqGHLjYSdaWcaaqaaiGbcYgaSjabc6gaUjabik% daYaqaaiabd2eannaaBaaaleaacqaIXaqmaeqaaOGaeiikaGIaem4s% aSKaeiykaKcaaaaa!4EFC! $$\psi (K) \geqslant \frac{{\ln 2}}{{M_1 (K)}}$$ , and give other upper and lower bounds. We conjecture that our upper bound is the exact value up to a constant. Our main tool is a general “Localization Lemma” that reduces integral inequalities over then-dimensional space to integral inequalities in a single variable. This lemma was first 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.