Relative entropies for convex bodies
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- by Justin Jenkinson and Elisabeth M. Werner PDF
- Trans. Amer. Math. Soc. 366 (2014), 2889-2906 Request permission
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
- MR Author ID: 252029
- ORCID: 0000-0001-9602-2172
- Email: elisabeth.werner@case.edu
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3180734