{"paper":{"title":"On the functional Minkowski problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.FA"],"primary_cat":"math.MG","authors_text":"Liran Rotem, Tomer Falah","submitted_at":"2025-02-24T07:43:50Z","abstract_excerpt":"To every log-concave function $f$ one may associate a pair of measures $(\\mu_{f},\\nu_{f})$ which are the surface area measures of $f$. These are a functional extension of the classical surface area measure of a convex body, and measure how the integral $\\int f$ changes under perturbations. The functional Minkowski problem then asks which pairs of measures can be obtained as the surface area measures of a log-concave function. In this work we fully solve this problem.\n  Furthermore, we prove that the surface area measures are continuous in correct topology: If $f_{k}\\to f$, then $\\left(\\mu_{f_{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.16929","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/2502.16929/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"}