{"paper":{"title":"On Continuous Terminal Embeddings of Sets of Positive Reach","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Mark Iwen, Rafael Chiclana, Simone Brugiapaglia, Tim Hoheisel","submitted_at":"2024-08-05T20:26:19Z","abstract_excerpt":"In this paper we prove the existence of H\\\"{o}lder continuous terminal embeddings of any desired $X \\subseteq \\mathbb{R}^d$ into $\\mathbb{R}^{m}$ with $m=\\mathcal{O}(\\varepsilon^{-2}\\omega(S_X)^2)$, for arbitrarily small distortion $\\varepsilon$, where $\\omega(S_X)$ denotes the Gaussian width of the unit secants of $X$. More specifically, when $X$ is a finite set we provide terminal embeddings that are locally $\\frac{1}{2}$-H\\\"{o}lder almost everywhere, and when $X$ is infinite with positive reach we give terminal embeddings that are locally $\\frac{1}{4}$-H\\\"{o}lder everywhere sufficiently clo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.02812","kind":"arxiv","version":2},"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/2408.02812/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"}