{"paper":{"title":"Tangency properties of sets with finite geometric curvature energies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CA","authors_text":"Sebastian Scholtes","submitted_at":"2012-02-02T16:07:54Z","abstract_excerpt":"We investigate inverse thickness $1/\\Delta$ and the integral Menger curvature energies $\\mathcal{U}_{p}^{\\alpha}$, $\\mathcal{I}_{p}^{\\alpha}$ and $\\mathcal{M}_{p}^{\\alpha}$, to find that finite $1/\\Delta$ or $\\mathcal{U}_{p}^{\\alpha}$ implies the existence of an approximate $\\alpha$-tangent at all points of the set, when $p\\geq \\alpha$ and that finite $\\mathcal{I}_{p}^{\\alpha}$ or $\\mathcal{M}_{p}^{\\alpha}$ implies the existence of a weak approximate $\\alpha$-tangent at every point of the set for $p\\geq 2\\alpha$ or $p\\geq 3\\alpha$, respectively, if some additional density properties hold. This"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.0472","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"}