Sections and projections of the outer and inner regularizations of a convex body

TL;DR

This paper introduces new geometric inequalities relating the volumes of sections and projections of convex bodies with their regularizations, extending these results to log-concave functions using recent optimal estimates.

Contribution

It presents novel inequalities for convex bodies' sections and projections, incorporating their regularizations, and extends these results to the functional setting of log-concave functions.

Findings

New inequalities for convex bodies and their regularizations.

Functional extensions to log-concave functions.

Utilization of recent optimal M-estimates for isotropic convex bodies.

Abstract

We establish new geometric inequalities comparing the volumes of sections and projections of a convex body, whose barycenter or Santal\'o point is at the origin, with those of its inner and outer regularizations. We also provide functional extensions of these inequalities to the setting of log-concave functions. Our approach relies on the recent optimal $M$-estimate of Bizeul and Klartag for isotropic convex bodies.