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Our main result reads that $\\mathscr{L}$ is not a locally-analytic"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1707.08288","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2017-07-26T05:03:56Z","cross_cats_sorted":[],"title_canon_sha256":"46ca6ef60f0e993ed8571e7eaa2d0d42857de81288c0e72c986b4b20bf37ed3a","abstract_canon_sha256":"27973b62888234c85a4c0a709be1bd24fd514c51c162f42d9fe2b809b12538bf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:18:31.685131Z","signature_b64":"mDKObfgYgd27oSCHAQkbJP23DpIslhmWznvhOQ02TYRhv9hJrwyosIOY37fK/mH54oLywDSUB3bjPiO4dRQTCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"29035c6e7e619c0404f491382c9c24a430f06a319195025dbb89db5f0befc4c7","last_reissued_at":"2026-05-18T00:18:31.684579Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:18:31.684579Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $k=1,\\dots,5$. 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