Functional affine-isoperimetry and an inverse logarithmic Sobolev inequality

TL;DR

This paper introduces a functional affine isoperimetric inequality for log-concave functions, providing an inverse form of the logarithmic Sobolev inequality related to entropy, with implications for Gaussian measures.

Contribution

It presents a novel functional affine isoperimetric inequality and its linearization, offering an inverse perspective on classical inequalities for log-concave functions and Gaussian measures.

Findings

Established a functional affine isoperimetric inequality for log-concave functions.

Derived an inverse logarithmic Sobolev inequality related to entropy.

Provided an inverse Poincaré inequality for Gaussian measures.

Abstract

We give a functional version of the affine isoperimetric inequality for log-concave functions which may be interpreted as an inverse form of a logarithmic Sobolev inequality inequality for entropy. A linearization of this inequality gives an inverse inequality to the Poincar'e inequality for the Gaussian measure.