Entropy, slicing problem and functional Mahler's conjecture

TL;DR

This paper extends geometric results related to the slicing and Mahler's conjectures to the setting of log-concave functions, using log-Laplace perturbations and entropy considerations to explore their equivalences.

Contribution

It introduces a functional framework for the slicing and Mahler's conjectures, incorporating entropy into the isotropic constant and analyzing functional analogues of geometric perturbations.

Findings

Functional analogues of geometric perturbations are characterized by log-Laplace transforms.

Entropy plays a crucial role in defining the suitable isotropic constant for functions.

Equivalences between functional and geometric strong forms of the slicing conjecture are analyzed.

Abstract

In a recent work, Bo'az Klartag showed that, given a convex body with minimal volume product, its isotropic constant is related to its volume product. As a consequence, he obtained that a strong version of the slicing conjecture implies Mahler's conjecture. In this work, we extend these geometrical results to the realm of log-concave functions. In this regard, the functional analogues of the projective perturbations of the body are the log-Laplace perturbations of the function. The differentiation along these transformations is simplified thanks to the known properties of the log-Laplace transform. Moreover, we show that achieving such an analogous result requires the consideration of the suitable version of the isotropic constant, notably the one incorporating the entropy. Finally, an investigation into the equivalences between the functional and geometrical strong forms of the slicing conjecture is provided.