Dual mixed volumes

Abstract

A concept dual to the mixed volumes of Minkowski is introduced.Duals to the classical mixed volume inequalities of Minkowski, Fenchel and Aleksandrov are presented.As an application of this work a sharp isoperimetric inequality relating the mean width of a convex body and the cross-sectional measures of its polar body is obtained.This inequality implies that of all convex bodies of a given mean width the n-ball (centered at the origin) is the one whose polar body has minimal cross-sectional measures of any index.It further gives a sharp lower bound for the product of the mean widths of a convex body and its polar body.