{"paper":{"title":"Inverse Blaschke-Santal\\'o inequality for convex curves enclosing the origin several times","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"E. Makai Jr, K. J. B\\\"or\\\"oczky","submitted_at":"2019-05-28T12:26:39Z","abstract_excerpt":"H. Guggenheimer generalized the planar volume product problem for locally convex curves $C$ enclosing the origin $k \\ge 2$ times. He conjectured that the minimal volume product $V(C)V(C^*)$ for these curves is attained if the curve consists of the longest diagonals of a regular $(2k+1)$-gon, with centre $0$, these diagonals taken always in the positive orientation. This conjectured minimum is of the form $k^2 + O(k)$. We investigate special cases of this conjecture. We prove it for locally convex $n$-gons with $2k+1 \\le n \\le 4k$, if the central angles at $0$ of all sides are equal to $2k \\pi "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.11766","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"}