{"paper":{"title":"The Riesz $\\alpha$-energy of log-concave functions and related Minkowski problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MG"],"primary_cat":"math.FA","authors_text":"Deping Ye, Niufa Fang, Zengle Zhang","submitted_at":"2024-08-28T21:07:37Z","abstract_excerpt":"We calculate the first order variation of the Riesz $\\alpha$-energy of a log-concave function $f$ with respect to the Asplund sum. Such a variational formula induces the Riesz $\\alpha$-energy measure of log-concave function $f$, which will be denoted by $\\mathfrak{R}_{\\alpha}(f, \\cdot)$. We pose the related Riesz $\\alpha$-energy Minkowski problem aiming to find necessary and/or sufficient conditions on a pregiven Borel measure $\\mu$ defined on $\\Rn$ so that $\\mu=\\mathfrak{R}_{\\alpha}(f,\\cdot)$ for some log-concave function $f$. Assuming enough smoothness, the Riesz $\\alpha$-energy Minkowski pr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.16141","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2408.16141/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}