# Geometric stability via information theory

@article{Ellis2015GeometricSV, title={Geometric stability via information theory}, author={David Ellis and Ehud Friedgut and Guy Kindler and Amir Yehudayoff}, journal={ArXiv}, year={2015}, volume={abs/1510.00258} }

The Loomis-Whitney inequality, and the more general Uniform Cover inequality, bound the volume of a body in terms of a product of the volumes of lower-dimensional 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…

## 14 Citations

Measure concentration and the weak Pinsker property

- Mathematics, Computer Science
- 2017

It is proved that μ may be represented as a mixture of other measures in which most of the weight in the mixture is on measures that exhibit a strong kind of concentration, and the number of summands is bounded in terms of the difference between the Shannon entropy of μ$\mu$ and the combined Shannon entropies of its marginals.

Inequalities on Projected Volumes

- MathematicsSIAM J. Discret. Math.
- 2021

It is proved that the closed convex hull $\overline{\conv}(\psi_n)$ is equal to the cone given by the uniform cover inequalities, however, perhaps surprisingly, it is shown that $\conv (\psi-n) $ is not closed for $n\ge 4$, thus disproving the conjecture.

A sharp Freiman type estimate for semisums in two and three dimensional Euclidean spaces

- Mathematics
- 2019

Freiman's Theorem is a classical result in additive combinatorics concerning the approximate structure of sets of integers that contain a high proportion of their internal sums. As a consequence, one…

On the nonlinear Brascamp–Lieb inequality

- Mathematics
- 2018

We prove a nonlinear variant of the general Brascamp-Lieb inequality. Instances of this inequality are quite prevalent in analysis, and we illustrate this with substantial applications in harmonic…

Higher order transversality in harmonic analysis

- Mathematics
- 2022

In differential topology two smooth submanifolds S1 and S2 of euclidean space are said to be transverse if the tangent spaces at each common point together form a spanning set. The purpose of this…

A proof of a Loomis–Whitney type inequality via optimal transport

- Mathematics, EconomicsJournal of Mathematical Analysis and Applications
- 2019

Measures of correlation and mixtures of product measures

- Computer Science, MathematicsArXiv
- 2018

The first part of this paper sets up the theory of TC and DTC for general random variables, not necessarily finite-valued, and considers the structural implications when a joint distribution $\mu$ has small TC or DTC.

Multi-variate correlation and mixtures of product measures

- Computer ScienceKybernetika
- 2020

The first part of this paper sets up the theory of TC and DTC for general random variables, not necessarily finite-valued, and considers the structural implications when a joint distribution $\mu$ has small TC or DTC.

On the Reverse Loomis–Whitney Inequality

- MathematicsDiscret. Comput. Geom.
- 2018

Some structural results are proved, bounds are given on the supremum of all δ, and the problem of actually computing the LW-constant of a rational polytope is dealt with.

Network nonlocality via rigidity of token counting and color matching

- Computer SciencePhysical Review A
- 2022

This paper introduces two families of strategies to produce nonlocal correlations in networks and shows that TC and CM distributions are rigidity in wide classes of networks, meaning that there is essentially a unique classical strategy to simulate such correlations.

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