{"paper":{"title":"Gr\\\"unbaum's inequality for sections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Matthew Stephen, Ning Zhang, Sergii Myroshnychenko","submitted_at":"2017-11-03T02:18:32Z","abstract_excerpt":"We show \\begin{align*} \\frac{ \\int_{E \\cap \\theta^+} f(x) dx }{ \\int_E f(x) dx }\n  \\geq \\left(\\frac{k \\gamma+1}{(n+1) \\gamma+1}\\right)^{\\frac{k \\gamma+1}{\\gamma}} \\end{align*} for all $k$-dimensional subspaces $E\\subset\\mathbb{R}^n$, $\\theta\\in E\\cap S^{n-1}$, and all $\\gamma$-concave functions $f:\\mathbb{R}^n\\rightarrow [0,\\infty)$ with $\\gamma >0$, $0< \\int_{\\mathbb{R}^n} f(x)\\, dx <\\infty$, and $\\int_{\\mathbb{R}^n} x f(x)\\, dx$ at the origin $o\\in\\mathbb{R}^n$. Here, $\\theta^+ := \\lbrace x\\, : \\, \\langle x,\\theta\\rangle \\geq 0 \\rbrace$. As a consequence of this result, we get the following "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.00998","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"}