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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.