Mathematika

Research Article

Volume Inequalities for Lp-Zonotopes

Professor Stefano Campia1 and Professor Paolo Gronchia2

a1 Dipartimento di Ingegneria dell'Informazione, Università degli Studi di Siena, Via Roma 56, 53100 Siena, Italy. E-mail: campi@dii.unisi.it

a2 Dipartimento di Matematica e Applicazioni per l'Architettura, Università degli Studi di Firenze, Piazza Ghiberti 27, 50122 Firenze, Italy. E-mail: paolo@fi.iac.cnr.it

Abstract

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 pth power of the support functions. An Lp-zonotope Zp is the p-sum of finitely many segments and is isometric to the unit ball of a subspace of ℓq, where 1/p + 1/q = 1. In this paper, a sharp upper estimate is given 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. In particular, for p = 1, the latter result provides a new approach to Reisner's inequality for the Mahler conjecture in the class of zonoids.

(Received February 06 2006)

MSC (2000)

  • Primary, 52A40

Footnotes

To Rolf Schneider, with gratitude.