The Brascamp-Lieb inequality in Convex Geometry and in the Theory of Algorithms

TL;DR

This paper explores the Brascamp-Lieb inequality's applications in convex and discrete geometry, its optimal constants, and its connections to algorithmic problems involving positive definite matrices.

Contribution

It provides a comprehensive analysis of the inequality's role in geometry and algorithms, highlighting new insights into optimal constants and extremizers.

Findings

The inequality's optimal constant can be characterized via Gaussian distributions.

Connections between the inequality and positive definite matrices are established.

Applications in convex and discrete geometry are demonstrated.

Abstract

The Brascamp-Lieb inequality in harmonic analysis was proved by Brascamp and Lieb in the rank one case in 1976, and by Lieb in 1990. It says that in a certain inequality, the optimal constant can be determined by checking the inequality for centered Gaussian distributions. It was Keith M Ball's pioneering work around 1990 that led to various applications of the inequality in Convex Geometry, and even in Discrete Geometry, like Brazitikos' quantitative fractional version of the Helly Theorem. On the other hand, determining the optimal constant and possible Gaussian extremizers for the Brascamp-Lieb inequality can be formulated as a problem in terms of positive definite matrices, and this problem has intimate links to the Theory of Algorithms.