{"paper":{"title":"R\\'enyi Divergence and $L_p$-affine surface area for convex bodies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Elisabeth M. Werner","submitted_at":"2011-05-05T17:31:54Z","abstract_excerpt":"We show that the fundamental objects of the $L_p$-Brunn-Minkowski theory, namely the $L_p$-affine surface areas for a convex body, are closely related to information theory: they are exponentials of R\\'enyi divergences of the cone measures of a convex body and its polar.\n  We give geometric interpretations for all R\\'enyi divergences $D_\\alpha$, not just for the previously treated special case of relative entropy which is the case $\\alpha =1$. Now, no symmetry assumptions are needed and, if at all, only very weak regularity assumptions are required.\n  Previously, the relative entropies appeare"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.1124","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}