Concavity principles for weighted marginals

TL;DR

This paper introduces a new framework for analyzing the concavity of weighted marginals of b2-concave functions, leading to functional inequalities like a dimensional Brunn-Minkowski and a Pre9kopa-type principle for rotationally invariant measures.

Contribution

It develops a general local method framework to study concavity properties, resulting in new functional inequalities for b2-concave and log-concave functions with rotational invariance.

Findings

Derived a functional dimensional Brunn-Minkowski inequality.

Established a Pre9kopa-type concavity principle for even log-concave functions.

Unified concavity principles under rotational invariance.

Abstract

We develop a general framework to study concavity properties of weighted marginals of $\beta$-concave functions on $\mathbb{R}^n$ via local methods. As a concrete implementation of our approach, we obtain a functional version of the dimensional Brunn-Minkowski inequality for rotationally invariant log-concave measures. Moreover, we derive a Pr\'ekopa-type concavity principle with rotationally invariant weights for even log-concave functions which encompasses the B-inequality.