{"paper":{"title":"Simultaneous smoothness and simultaneous stability of a $C^\\infty$ strictly convex integrand and its dual","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Erica Boizan Batista, Huhe Han, Takashi Nishimura","submitted_at":"2017-07-06T11:33:34Z","abstract_excerpt":"In this paper, we investigate simultaneous properties of a convex integrand $\\gamma$ and its dual $\\delta$. The main results are the following three.\n  (1) For a $C^\\infty$ convex integrand $\\gamma: S^n\\to \\mathbb{R}_+$, its dual convex integrand $\\delta: S^n\\to \\mathbb{R}_+$ is of class $C^\\infty$ if and only if $\\gamma$ is a strictly convex integrand.\n  (2) Let $\\gamma: S^n\\to \\mathbb{R}_+$ be a $C^\\infty$ strictly convex integrand. Then, $\\gamma$ is stable if and only if its dual convex integrand $\\delta: S^n\\to \\mathbb{R}_+$ is stable.\n  (3) Let $\\gamma: S^n\\to \\mathbb{R}_+$ be a $C^\\infty"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.02359","kind":"arxiv","version":1},"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"}