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.
