{"paper":{"title":"Nearly sharp comparison results for sliced and max-sliced Wasserstein distances","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.FA","math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Jacob Shkrob, Jonathan Niles-Weed","submitted_at":"2026-08-13T15:36:23Z","abstract_excerpt":"We prove new comparison results between the Wasserstein distance and its sliced and max-sliced counterparts. First, we show that the H\\\"older exponent~$\\frac{2}{d+2}$ obtained by Bobkov and G\\\"otze for the max-sliced 1-Wasserstein distance on the unit ball is optimal for every $d \\geq 2$, settling a question raised in their work. Second, we show that sharper comparisons are possible under stronger structural assumptions: if $\\nu$ is a discrete measure and the optimal coupling between $\\mu$ and $\\nu$ transports each point to a nearest atom of $\\nu$, then $W_p(\\mu, \\nu) \\leq C \\sqrt{d}\\, K \\, \\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.13374","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/2608.13374/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"}