{"paper":{"title":"Stability in the Busemann-Petty and Shephard problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Alexander Koldobsky","submitted_at":"2011-01-19T02:38:31Z","abstract_excerpt":"A comparison problem for volumes of convex bodies asks whether inequalities $f_K(\\xi)\\le f_L(\\xi)$ for all $\\xi\\in S^{n-1}$ imply that $\\vol_n(K)\\le \\vol_n(L),$ where $K,L$ are convex bodies in $\\R^n,$ and $f_K$ is a certain geometric characteristic of $K.$ By linear stability in comparison problems we mean that there exists a constant $c$ such that for every $\\e>0$, the inequalities $f_K(\\xi)\\le f_L(\\xi)+\\e$ for all $\\xi\\in S^{n-1}$ imply that $(\\vol_n(K))^{\\frac{n-1}n}\\le (\\vol_n(L))^{\\frac{n-1}n}+c\\e.$\n  We prove such results in the settings of the Busemann-Petty and Shephard problems and t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.3600","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"}