{"paper":{"title":"Why there is no an existence theorem for a convex polytope with prescribed directions and perimeters of the faces?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Victor Alexandrov","submitted_at":"2017-07-26T05:03:56Z","abstract_excerpt":"We choose some special unit vectors $\\boldsymbol{n}_1,\\dots,\\boldsymbol{n}_5$ in $\\mathbb{R}^3$ and denote by $\\mathscr{L}\\subset\\mathbb{R}^5$ the set of all points $(L_1,\\dots,L_5)\\in\\mathbb{R}^5$ with the following property: there exists a compact convex polytope $P\\subset\\mathbb{R}^3$ such that the vectors $\\boldsymbol{n}_1,\\dots,\\boldsymbol{n}_5$ (and no other vector) are unit outward normals to the faces of $P$ and the perimeter of the face with the outward normal $\\boldsymbol{n}_k$ is equal to $L_k$ for all $k=1,\\dots,5$. Our main result reads that $\\mathscr{L}$ is not a locally-analytic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.08288","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"}