Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On volume product inequalities for convex sets

Authors: Stefano Campi and Paolo Gronchi
Journal: Proc. Amer. Math. Soc. 134 (2006), 2393-2402
MSC (2000): Primary 52A40
Published electronically: February 3, 2006
MathSciNet review: 2213713
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Abstract: The volume of the polar body of a symmetric convex set $ K$ of $ {\mathbb{R}^d}$ is investigated. It is shown that its reciprocal is a convex function of the time $ t$ along movements, in which every point of $ K$ moves with constant speed parallel to a fixed direction.

This result is applied to find reverse forms of the $ L^{p}$-Blaschke-Santaló inequality for two-dimensional convex sets.

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Additional Information

Stefano Campi
Affiliation: Dipartimento di Matematica Pura e Applicata “G. Vitali", Università degli Studi di Modena e Reggio Emilia, Via Campi 213/b, 41100 Modena, Italy
Address at time of publication: Dipartimento di Ingegneria dell’Informazione, Università degli Studi di Siena, Via Roma 56, 53100 Siena, Italy

Paolo Gronchi
Affiliation: Istituto per le Applicazioni del Calcolo - Sezione di Firenze, Consiglio Nazionale delle Ricerche Via Madonna del Piano, Edificio F, 50019 Sesto Fiorentino (FI), Italy
Address at time of publication: Dipartimento di Matematica e Applicazioni per l’Architettura, Università degli Studi di Firenze, Piazza Ghiberti 27, 50122 Firenze, Italy

Keywords: Polar body, volume product, $L^{p}$-centroid bodies
Received by editor(s): July 27, 2004
Received by editor(s) in revised form: March 4, 2005
Published electronically: February 3, 2006
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.