{"paper":{"title":"Sharp Poincar\\'e Interpolation Along Wasserstein Geodesics","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.FA","math.MG","math.SP"],"primary_cat":"math.PR","authors_text":"Bang-Xian Han, Zhuo-Nan Zhu","submitted_at":"2026-07-12T13:51:42Z","abstract_excerpt":"We prove a sharp interpolation inequality for the Poincar\\'e constant along quadratic Wasserstein geodesics. Let $\\mu_i$, $i=0,1$, be $\\kappa_i$-strongly log-concave probability measures on $\\mathbb R^n$, and let $(\\mu_t)_{t\\in[0,1]}$ be their optimal displacement interpolation. Then \\[ \\sqrt{C_P(\\mu_t)} \\leq \\frac{1-t}{\\sqrt{\\kappa_0}} + \\frac{t}{\\sqrt{\\kappa_1}}. \\] This estimate is optimal for every $t,\\kappa_0,\\kappa_1$, holds for all test functions without symmetry assumptions, and remains valid for extended-valued potentials.\n  We also characterize equality at an interior time: it holds "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.10769","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/2607.10769/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"}