{"paper":{"title":"Proof of the Holevo--Utkin conjecture on sharp $\\ell_p$ norms for zero-sum vectors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The conjectured sharp bounds on p-norm to 2-norm ratios for zero-sum vectors hold for all dimensions four and higher.","cross_cats":["math-ph","math.FA","math.MP"],"primary_cat":"math.CA","authors_text":"Haonan Zhang","submitted_at":"2026-05-04T03:11:54Z","abstract_excerpt":"Let $d\\ge 3$ and $p>0$. Let $\\|x\\|_p$ denote the $\\ell_p$ (quasi-)norm of a $d$-dimensional vector $x$. Holevo and Utkin \\cite{HU26} conjectured that for $0<p\\le 1$, \\[ \\min \\left\\{\\frac{\\|x\\|_p}{\\|x\\|_2}:\\vec{0}\\neq x\\in\\mathbb R^d,\\ \\sum_{i=1}^d x_i=0\\right\\} =2^{1/p-1/2}; \\] for $1<p<2$, \\[ \\min \\left\\{\\frac{\\|x\\|_p}{\\|x\\|_2}:\\vec{0}\\neq x\\in\\mathbb R^d,\\ \\sum_{i=1}^d x_i=0\\right\\} = \\min\\left\\{2^{1/p-1/2},\\left(\\frac{(d-1)^{p/2}+(d-1)^{1-p/2}}{d^{p/2}}\\right)^{1/p}\\right\\}; \\] and for $2<q<\\infty$ \\[ \\max\\left\\{\\frac{\\|x\\|_q}{\\|x\\|_2}:\\vec{0}\\neq x\\in\\mathbb R^d,\\ \\sum_{i=1}^d x_i=0\\right\\"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For d ≥ 4 the minimum of ||x||_p / ||x||_2 over non-zero zero-sum x equals 2^{1/p-1/2} when 0 < p ≤ 1, equals the min of that quantity and ((d-1)^{p/2} + (d-1)^{1-p/2})/d^{p/2})^{1/p} when 1 < p < 2, and the analogous maximum statement holds for q > 2.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The proof must correctly identify and compare the two candidate extremal configurations (the two-support vector and the equitable (d-1)-support vector) and show no other zero-sum vector yields a smaller or larger ratio; this case analysis is not visible in the abstract.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Proves that the minimum and maximum of ||x||_p / ||x||_2 over non-zero zero-sum x in R^d equal the stated closed-form expressions for all d ≥ 4.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The conjectured sharp bounds on p-norm to 2-norm ratios for zero-sum vectors hold for all dimensions four and higher.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"7ca6f1d23df7364dca4daf1199fff5fc51f9ab8e95c002953aee07cca90bba5c"},"source":{"id":"2605.05243","kind":"arxiv","version":2},"verdict":{"id":"e8d2d0ba-8e2b-43ad-9a2b-10705592c9a9","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T17:19:21.557797Z","strongest_claim":"For d ≥ 4 the minimum of ||x||_p / ||x||_2 over non-zero zero-sum x equals 2^{1/p-1/2} when 0 < p ≤ 1, equals the min of that quantity and ((d-1)^{p/2} + (d-1)^{1-p/2})/d^{p/2})^{1/p} when 1 < p < 2, and the analogous maximum statement holds for q > 2.","one_line_summary":"Proves that the minimum and maximum of ||x||_p / ||x||_2 over non-zero zero-sum x in R^d equal the stated closed-form expressions for all d ≥ 4.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The proof must correctly identify and compare the two candidate extremal configurations (the two-support vector and the equitable (d-1)-support vector) and show no other zero-sum vector yields a smaller or larger ratio; this case analysis is not visible in the abstract.","pith_extraction_headline":"The conjectured sharp bounds on p-norm to 2-norm ratios for zero-sum vectors hold for all dimensions four and higher."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.05243/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T16:36:42.607288Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-20T04:01:22.356213Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T16:37:36.391377Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"206b1be5a2e01f818ab810a65da1ce0888d2935ff7ca09524370efedd90346ad"},"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"}