Average mixed volume under projection

TL;DR

This paper establishes an upper bound on the average mixed volume of projected convex bodies in high-dimensional space, linking it to a specific quermassintegral, thus advancing understanding of geometric properties under projection.

Contribution

It introduces a new bound relating average mixed volume under projection to quermassintegrals, providing a novel geometric inequality for convex bodies.

Findings

Bound on average mixed volume in terms of quermassintegral

Extension of mixed volume inequalities to projections

Insights into geometric behavior of convex bodies under random projections

Abstract

The average mixed volume of a random projection of $d$ convex bodies in $\mathbb R^n$ is bounded above in terms of a quermassintegral.