A note on Bourgain-Milman's universal constant

TL;DR

This paper investigates lower bounds on the volume product of convex bodies in high-dimensional spaces using a modified logarithmic Brunn-Minkowski inequality, contributing to the understanding of Bourgain-Milman's universal constant.

Contribution

It introduces a new approach to estimate volume products for convex bodies without symmetry assumptions via a modified inequality.

Findings

Established lower estimates on volume products in $R^n$

Connected volume bounds to the logarithmic Brunn-Minkowski inequality

Provided insights into Bourgain-Milman's universal constant

Abstract

The present note is a result of an on-going investigation into the logarithmic Brunn-Minkowski inequality. We obtain lower estimates on the volume product for convex bodies in $\mathbb{R}^n$ not necessarily symmetric with respect to the origin from a modified logarithmic Brunn-Minkowski inequality.