On successive minima-type inequalities for the polar of a convex body

TL;DR

This paper investigates inequalities relating the volume of a convex body to the successive minima of its polar, motivated by longstanding conjectures in convex geometry.

Contribution

It provides new bounds on convex body volume based on the successive minima of its polar, advancing understanding of Mahler and Makai Jr.'s conjectures.

Findings

Derived new inequalities linking volume and successive minima

Provided bounds that improve or extend previous results

Contributed to the theoretical framework of convex geometry

Abstract

Motivated by conjectures of Mahler and Makai Jr., we study bounds on the volume of a convex body in terms of the successive minima of its polar body.