Geometric stability via information theory
Abstract
The LoomisWhitney inequality, and the more general Uniform Cover inequality, bound the volume of a body in terms of a product of the volumes of lowerdimensional projections of the body. In this paper, we prove stability versions of these inequalities, showing that when they are close to being tight, the body in question is close in symmetric difference to a 'box'. Our results are best possible up to a constant factor depending upon the dimension alone. Our approach is information theoretic. We use our stability result for the LoomisWhitney inequality to obtain a stability result for the edgeisoperimetric inequality in the infinite $d$dimensional lattice. Namely, we prove that a subset of $\mathbb{Z}^d$ with small edgeboundary must be close in symmetric difference to a $d$dimensional cube. Our bound is, again, best possible up to a constant factor depending upon $d$ alone.
 Publication:

arXiv eprints
 Pub Date:
 September 2015
 arXiv:
 arXiv:1510.00258
 Bibcode:
 2015arXiv151000258E
 Keywords:

 Mathematics  Metric Geometry;
 Computer Science  Information Theory;
 Mathematics  Combinatorics;
 52C07;
 05D99;
 G.2.1
 EPrint:
 28 pages. Reformatted for Discrete Analysis, but otherwise identical to the previous version