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Stability of the reverse Blaschke-Santaló inequality for unconditional convex bodies


Authors: Jaegil Kim and Artem Zvavitch
Journal: Proc. Amer. Math. Soc. 143 (2015), 1705-1717
MSC (2010): Primary 52A20, 53A15, 52B10
Published electronically: November 20, 2014
MathSciNet review: 3314083
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Abstract: Mahler's conjecture asks whether the cube is a minimizer for the volume product of a body and its polar in the class of symmetric convex bodies in $ \mathbb{R}^n$. The inequality corresponding to the conjecture is sometimes called the reverse Blaschke-Santaló inequality. The conjecture is known to be true in $ \mathbb{R}^2$ and for several special cases. In the class of unconditional convex bodies, Saint-Raymond confirmed the conjecture, and Meyer and Reisner, independently, characterized the equality case. In this paper we present a stability version of these results and also show that any symmetric convex body, which is sufficiently close to an unconditional body, satisfies the reverse Blaschke-Santaló inequality.


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Additional Information

Jaegil Kim
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada
Email: jaegil@ualberta.ca

Artem Zvavitch
Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242
Email: zvavitch@math.kent.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12334-3
Keywords: Convex bodies, polar bodies, unconditional convex bodies, Hanner polytopes, volume product, Mahler's conjecture, Blaschke-Santal\'o inequality.
Received by editor(s): March 2, 2013
Received by editor(s) in revised form: August 12, 2013, and August 15, 2013
Published electronically: November 20, 2014
Additional Notes: The authors were supported in part by U.S. National Science Foundation grant DMS-1101636, and the first author is also supported in part by NSERC
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.