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Relative entropies for convex bodies


Authors: Justin Jenkinson and Elisabeth M. Werner
Journal: Trans. Amer. Math. Soc. 366 (2014), 2889-2906
MSC (2010): Primary 52A20, 53A15
DOI: https://doi.org/10.1090/S0002-9947-2014-05788-7
Published electronically: February 6, 2014
MathSciNet review: 3180734
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Abstract: We introduce a new class of (not necessarily convex) bodies and show, among other things, that these bodies provide yet another link between convex geometric analysis and information theory. Namely, they give geometric interpretations of the relative entropy of the cone measures of a convex body and its polar and related quantities.

Such interpretations were first given by Paouris and Werner for symmetric convex bodies in the context of the $ L_p$-centroid bodies. There, the relative entropies appear after performing second order expansions of certain expressions. Now, no symmetry assumptions are needed. Moreover, using the new bodies, already first order expansions make the relative entropies appear. Thus, these bodies detect ``faster'' details of the boundary of a convex body than the $ L_p$-centroid bodies.


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Additional Information

Justin Jenkinson
Affiliation: Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106
Email: jdj13@case.edu

Elisabeth M. Werner
Affiliation: Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106 – and – UFR de Mathématique, Université de Lille 1, 59655 Villeneuve d’Ascq, France
Email: elisabeth.werner@case.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-05788-7
Keywords: Relative entropy, mean width, $L_p$-affine surface area
Received by editor(s): June 15, 2011
Received by editor(s) in revised form: December 9, 2011
Published electronically: February 6, 2014
Additional Notes: This work was partially supported by an NSF grant, an FRG-NSF grant and a BSF grant
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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