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. 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 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. 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.
Volume inequalities for Lp-zonotopes / S. CAMPI; P. GRONCHI. - In: MATHEMATIKA. - ISSN 0025-5793. - STAMPA. - 53:(2006), pp. 71-80. [10.1112/S0025579300000036]
Volume inequalities for Lp-zonotopes
GRONCHI, PAOLO
2006
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 p-th 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 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. 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.File | Dimensione | Formato | |
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