Polar duals of convex bodies

the long version

A generalization and the dual version of the following result due to Firey is given: The mixed area of a plane convex body and its polar dual is at least Pi. We give a sharp upper bound for the product of the dual cross- sectional measure of any index and that of its polar dual. A general result for a convex body Kappa and a convex increasing real valued function gives inequalities for sets of constant width and sets with equichordal points as special cases. Keywords: Polar duals; Convex bodies. (JHD)