{"paper":{"title":"Applications of Gr\\\"unbaum-type inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Matthew Stephen, Vladyslav Yaskin","submitted_at":"2018-09-15T21:57:15Z","abstract_excerpt":"Let $1\\leq i \\leq k < n$ be integers. We prove the following exact inequalities for any convex body $K\\subset\\mathbb{R}^n$ with centroid at the origin, and any $k$-dimensional subspace $E\\subset \\mathbb{R}^n$: \\begin{align*} &V_i \\big( K\\cap E \\big) \\geq \\left( \\frac{i+1}{n+1} \\right)^i \\max_{x\\in K} V_i \\big( ( K-x) \\cap E \\big) , \\\\ &\\widetilde{V}_i \\big( K\\cap E \\big) \\geq \\left( \\frac{i+1}{n+1} \\right)^i \\max_{x\\in K} \\widetilde{V}_i \\big( ( K-x) \\cap E \\big) ; \\end{align*} $V_i$ is the $i$th intrinsic volume, and $\\widetilde{V}_i$ is the $i$th dual volume taken within $E$. Our results are"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.05775","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"}