Functional analogs of the Shephard, Busemann-Petty, and Milman problems

TL;DR

This paper explores functional versions of classical convex geometry problems, providing bounds that generalize known results for convex sets, thereby extending the scope of these geometric inequalities.

Contribution

It introduces and analyzes functional analogs of the Shephard, Busemann-Petty, and Milman problems, offering new bounds and generalizations.

Findings

Established bounds for functional Shephard problem

Extended Busemann-Petty problem to functional setting

Generalized Milman problem bounds for functions

Abstract

The paper studies possible functional analogs of classical problems from convex geometry. In particular, we provide some bounds in the functional Shephard, Busemann-Petty, and Milman problems generalizing known bounds in this problems for convex sets.