TY - JOUR

T1 - Relative entropies for convex bodies

AU - Jenkinson, Justin

AU - Werner, Elisabeth M.

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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 Lp-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 Lp-centroid bodies.

AB - 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 Lp-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 Lp-centroid bodies.

KW - L-affine surface area

KW - Mean width

KW - Relative entropy

UR - http://www.scopus.com/inward/record.url?scp=84897725175&partnerID=8YFLogxK

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U2 - 10.1090/S0002-9947-2014-05788-7

DO - 10.1090/S0002-9947-2014-05788-7

M3 - Article

AN - SCOPUS:84897725175

VL - 366

SP - 2889

EP - 2906

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 6

ER -