Functional Liftings of Restricted Geometric Inequalities

TL;DR

This paper introduces generalized sup-convolutions to derive functional inequalities from geometric inequalities, extending classical results like Borell-Brascamp-Lieb and the log-Brunn-Minkowski conjecture to broader settings including Euclidean spaces and nilpotent Lie groups.

Contribution

It develops a framework connecting geometric inequalities with their functional counterparts via sup-convolution techniques, applicable in diverse mathematical contexts.

Findings

Derived a Borell-Brascamp-Lieb inequality for Gaussian spaces.

Established a functional analog of the log-Brunn-Minkowski conjecture.

Extended inequalities to nilpotent Lie groups.

Abstract

We investigate what we term "generalized sup-convolutions". We show that functional inequalities that enjoy an interpretation as sup-convolution inequalities can be deduced from the special case of indicator functions corresponding to a geometric inequality. As consequences we derive a Borell-Brascamp Lieb inequality for the Gaussian Brunn-Minkowski inequality and give a functional analog of the log-Brunn Minkowski conjecture. Though we focus on Euclidean applications, our results are general and can be directly applied in more abstract settings, like groups or even topological measure spaces without algebraic structure, we instantiate this claim with a Borell-Brascamp-Lieb type inequality for nilpotent Lie groups.