{"paper":{"title":"Stability and slicing inequalities for intersection bodies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Alexander Koldobsky, Dan Ma","submitted_at":"2011-08-12T14:40:14Z","abstract_excerpt":"We prove a generalization of the hyperplane inequality for intersection bodies, where volume is replaced by an arbitrary measure $\\mu$ with even continuous density and sections are of arbitrary dimension $n-k,\\ 1\\le k <n.$ If $K$ is a generalized $k$-intersection body, then $$\\mu(K)\\,\\leq\\,\\frac{n}{n-k}c_{n,k}\\max_{H} \\mu(K\\cap H) \\vol_n(K)^{k/n}.$$ Here $c_{n,k} = |B_2^n|^{(n-k)/n}/|B_2^{n-k}|<1,$ $|B_2^n|$ is the volume of the unit Euclidean ball, and maximum is taken over all $(n-k)$-dimensional subspaces of $\\R^n.$ The constant is optimal, and for each intersection body the inequality hold"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.2631","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"}