{"paper":{"title":"Volume product of planar polar convex bodies --- lower estimates with stability","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, M. Meyer, S. Reisner","submitted_at":"2015-07-06T14:43:57Z","abstract_excerpt":"Let $K \\subset {\\mathbb R}^2$ be an $o$-symmetric convex body, and $K^*$ its polar body. Then we have $|K|\\cdot |K^*| \\ge 8$, with equality if and only if $K$ is a parallelogram. ($| \\cdot |$ denotes volume). If $K \\subset {\\mathbb R}^2$ is a convex body, with $o \\in {\\text{int}}\\,K$, then $|K|\\cdot |K^*| \\ge 27/4$, with equality if and only if $K$ is a triangle and $o$ is its centroid. If $K \\subset {\\mathbb R}^2$ is a convex body, then we have $|K| \\cdot |[(K-K)/2)]^* | \\ge 6$, with equality if and only if $K$ is a triangle. These theorems are due to Mahler and Reisner, Mahler and Meyer, and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.01481","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"}