A reverse Minkowski-type inequality

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.