Santalo region of a log-concave function

TL;DR

This paper introduces the concepts of Santalo region and Floating body for log-concave functions, exploring their properties and establishing a connection to convex bodies, thereby extending geometric relations to a functional setting.

Contribution

It defines and studies the Santalo region and Floating body for log-concave functions, linking these to convex body relations in a novel way.

Findings

Relation between Floating body and Santalo region of convex bodies extends to log-concave functions

Properties of Santalo region and Floating body for log-concave functions are characterized

Main result connects geometric relations in convex bodies to functional analogs

Abstract

In this paper we define the Santalo region and the Floating body of a log-concave function. We then study their properties. Our main result is that any relation of Floating body and Santalo region of a convex body is translated to a relation of Floating body and Santalo region of an even log-concave function