{"paper":{"title":"Viviani Polytopes and Fermat Points","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.HO"],"primary_cat":"math.MG","authors_text":"Li Zhou","submitted_at":"2010-08-06T16:57:41Z","abstract_excerpt":"Given a set of oriented hyperplanes $\\mathcal{P}=\\{p_1, ..., p_k\\}$ in $\\mathbb{R}^n$, define $v(P)$ for any point $P\\in\\mathbb{R}^n$ as the sum of the signed distances from $P$ to $p_1$,..., $p_k$. We give a simple geometric characterization of $\\mathcal{P}$ so that $v$ is a constant. The characterization leads to a connection with the Fermat point of $k$ points in $\\mathbb{R}^n$. Finally, we discuss historically the full content of Viviani's theorem."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.1236","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"}