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Takahasi","submitted_at":"2011-09-08T17:07:28Z","abstract_excerpt":"For a normed linear space $(X,|\\cdot|)$ and $p>0$ we characterize all $n$-tuples $(\\mu_1,...,\\mu_n)\\in\\mathbb{R}^{n}$ for which the generalized triangle inequality of the second type $$\\|x_1+...+x_n\\|^p\\leq\\frac{|x_1|^p}{\\mu_1}+...+\\frac{|x_n|^p}{\\mu_n}$$ holds for any $x_1,...,x_n\\in X$. We also characterize $(\\mu_1,...,\\mu_n)\\in\\mathbb{R}^{n}$ for which the reverse of the inequality above holds."},"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":"1109.1773","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-09-08T17:07:28Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"0c3183fb88de372f52da7d45c6fa560a51b93b95210600a82c9443c63f435088","abstract_canon_sha256":"10e8da2c33998dba2b6834eebef21ace797c598eaa49b7da121ac8aff65258ab"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:59:40.018011Z","signature_b64":"lTANY1ojIbmegwsBV5x3ix8SNn+eIJYnhvTmYoqV0GAfwjgq/K+jaIU8lTeizRuUW68xssKRoJ5M0JZ3jf/ABQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"69978354b8eda5b1542401b186243a5499fcdd3b33dd7808a2912bb9ac5e2323","last_reissued_at":"2026-05-18T03:59:40.017136Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:59:40.017136Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Characterization of a generalized triangle inequality in normed spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.FA","authors_text":"F. 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