A reverse Minkowski-type inequality

Abstract

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