Relative entropies for convex bodies

Justin Jenkinson, Elisabeth M. Werner

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)2889-2906
Number of pages18
JournalTransactions of the American Mathematical Society
Issue number6
StatePublished - 2014

Bibliographical note

Publisher Copyright:
© 2014 American Mathematical Society.


  • L-affine surface area
  • Mean width
  • Relative entropy


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