The Existence of Extremizers of Blaschke-Santal\'o Type Inequalities

TL;DR

This paper proves the existence of extremizers for the functional Santaló inequality and its reverse by analyzing the topological structure of convex functions and their polar transforms.

Contribution

It introduces a topological framework for convex functions and demonstrates the existence of extremizers for the Blaschke-Santaló type inequalities.

Findings

Existence of extremizers for the functional Santaló inequality.

Existence of extremizers for the reverse Santaló inequality.

Development of a topological structure on convex functions.

Abstract

We discuss a topological structure on families of convex functions and then apply it to show the existence of extrimizers for the functional Santal\'{o} inequality with respect to polar transform and its reverse.