{"paper":{"title":"Relative entropy of cone measures and $L_p$ centroid bodies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Elisabeth M. Werner, Grigoris Paouris","submitted_at":"2009-09-24T04:22:42Z","abstract_excerpt":"Let $K$ be a convex body in $\\mathbb R^n$. We introduce a new affine invariant, which we call $\\Omega_K$, that can be found in three different ways: as a limit of normalized $L_p$-affine surface areas, as the relative entropy of the cone measure of $K$ and the cone measure of $K^\\circ$, as the limit of the volume difference of $K$ and $L_p$-centroid bodies. We investigate properties of $\\Omega_K$ and of related new invariant quantities. In particular, we show new affine isoperimetric inequalities and we show a \"information inequality\" for convex bodies."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.4361","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"}