A remark about Mahler's conjecture and the maximum value of box splines

TL;DR

This paper links Mahler's conjecture to the maximum of box splines for polytopes with up to 2n+2 facets, providing an asymptotic formula and proving the conjecture for large n.

Contribution

It introduces a novel approach to Mahler's conjecture using box splines and proves the conjecture for large dimensions in this special case.

Findings

Asymptotic formula for univariate box splines.

Proof of Mahler's conjecture for polytopes with up to 2n+2 facets when n is large.

Connection established between Mahler's conjecture and maximum values of box splines.

Abstract

In this paper, we recast a special case of Mahler'c conjecture by the maximum value of box splines. This is the case of polytopes with at most $2n+2$ facets. An asymptotic formula for univariate box splines is given. Based on the formula, Mahler's conjecture is proved in this case provided $n$ is big enough.