THE VOLUME PRODUCT OF CONVEX BODIES WITH MANY HYPERPLANE SYMMETRIES

Mahler's conjecture predicts a sharp lower bound on the volume of the polar of a convex body in terms of its volume. We confirm the conjecture for convex bodies with many hyperplane symmetries in the following sense: their hyperplanes of symmetries have a one-point intersection. Moreover, we ob...

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Bibliographic Details
Published inAmerican journal of mathematics Vol. 135; no. 2; pp. 311 - 347
Main Authors Barthe, F., Fradelizi, M.
Format Journal Article
LanguageEnglish
Published Baltimore Johns Hopkins University Press 01.04.2013
Subjects
Functional Analysis
Geometry
Mathematical problems
Mathematics
Mathematics education
Polytopes
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Summary:Mahler's conjecture predicts a sharp lower bound on the volume of the polar of a convex body in terms of its volume. We confirm the conjecture for convex bodies with many hyperplane symmetries in the following sense: their hyperplanes of symmetries have a one-point intersection. Moreover, we obtain improved sharp lower bounds for classes of convex bodies which are invariant by certain reflection groups, namely direct products of the isometry groups of regular polytopes.
ISSN:0002-9327
1080-6377
1080-6377
DOI:10.1353/ajm.2013.0018