A reverse Minkowski-type inequality

TL;DR

This paper extends a known geometric inequality by establishing a sharp upper bound for mixed volume in higher dimensions, relating it to mean width and surface area, with a complete characterization of equality cases.

Contribution

It introduces a new sharp upper bound for mixed volume involving mean width and surface area, generalizing a planar result to higher dimensions.

Findings

Proves a sharp upper bound for mixed volume in terms of mean width and surface area.

Characterizes the equality cases completely.

Provides a stability improvement for related isoperimetric inequalities.

Abstract

The famous Minkowski inequality provides a sharp lower bound for the mixed volume $V(K,M[n-1])$ of two convex bodies $K,M\subset\mathbb{R}^n$ in terms of powers of the volumes of the individual bodies $K$ and $M$. The special case where $K$ is the unit ball yields the isoperimetric inequality. In the plane, Betke and Weil (1991) found a sharp upper bound for the mixed area of $K$ and $M$ in terms of the perimeters of $K$ and $M$. We extend this result to general dimensions by proving a sharp upper bound for the mixed volume $V(K,M[n-1])$ in terms of the mean width of $K$ and the surface area of $M$. The equality case is completely characterized. In addition, we establish a stability improvement of this and related geometric inequalities of isoperimetric type.