Geometric stability via information theory (Q2826226)

Stable versions of the Loomis-Whitney inequality and of the Uniform cover inequality as well as of other results, such as an edge-isoperimetric inequality in the lattice \(\mathbb{L}^d\) are provided by means of an information theoretic approach, showing that when they are close to being tight, the body in question is close in symmetric difference to a box. The authors show that their results are best possible up to constant factors depending solely on the dimension. A comprehensive review of the existing literature on the matter is provided in the preliminary section and the paper closes with two conjectures regarding the dependence on the dimension of some of the results.