Volume inequalities for Lp-zonotopes

The classical Minkowski sum of convex sets is defined by the sum of the corresponding support functions. The Lp extension of such a definition makes use of the sum of the p-th power of the support functions. A p-zonotope Zp is the p-sum of finitely many segments and is isometric to the unit ball of lq . In this paper we give a sharp upper estimate of the volume of Zp in terms of the volume of Z1 as well as a sharp lower estimate of the volume of the polar of Zp in terms of the same quantity. For p=1 the latter result provides a new approach to Reisner's inequality for the Mahler conjecture in the class of zonoids.