{"paper":{"title":"On sets in ${\\mathbb R}^d$ with DC distance function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Du\\v{s}an Pokorn\\'y, Lud\\v{e}k Zaj\\'i\\v{c}ek","submitted_at":"2019-04-27T22:56:53Z","abstract_excerpt":"We study closed sets $F \\subset {\\mathbb R}^d$ whose distance function $d_F:= {\\rm dist}\\,(\\cdot,F)$ is DC (i.e., is the difference of two convex functions on ${\\mathbb R}^d$). Our main result asserts that if $F \\subset {\\mathbb R}^2$ is a graph of a DC function $g:{\\mathbb R}\\to {\\mathbb R}$, then $F$ has the above property. If $d>1$, the same holds if $g:{\\mathbb R}^{d-1}\\to {\\mathbb R}$ is semiconcave, however the case of a general DC function $g$ remains open."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.12223","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":""},"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"}