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Shadow Systems and Volumes of Polar Convex Bodies

Part of: General convexity

Published online by Cambridge University Press:  21 December 2009

Mathieu Meyer  and
Shlomo Reisner
Mathieu Meyer
Affiliation:
Equipe d'Analyse et de Mathématiques Appliquées, Université de Marne-la-Vallée, Cité Descartes, 5, boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France. E-mail: mathieu.meyer@univ-mlv.fr
Shlomo Reisner
Affiliation:
Department of Mathematics, University of Haifa, Haifa 31905, Israel. E-mail: reisner@math.haifa.ac.il
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Abstract

It is proved that the reciprocal of the volume of the polar bodies, about the Santaló point, of a shadow system of convex bodies Kt, is a convex function of t, thus extending to the non-symmetric case a result of Campi and Gronchi. The case that the reciprocal of the volume is an affine function of t is also investigated and is characterized under certain conditions. These results are applied to prove an exact reverse Santaló inequality for polytopes in ℝd that have at most d + 3 vertices.

MSC classification

Secondary: 52A40: Inequalities and extremum problems
Type
Research Article
Information
Mathematika , Volume 53 , Issue 1 , June 2006 , pp. 129 - 148
Copyright
Copyright © University College London 2006

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