{"paper":{"title":"Affine quermassintegrals of random polytopes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.PR"],"primary_cat":"math.MG","authors_text":"Giorgos Chasapis, Nikos Skarmogiannis","submitted_at":"2019-06-19T10:34:10Z","abstract_excerpt":"A question related to some conjectures of Lutwak about the affine quermassintegrals of a convex body $K$ in ${\\mathbb R}^n$ asks whether for every convex body $K$ in ${\\mathbb R}^n$ and all $1\\leqslant k\\leqslant n$ $$\\Phi_{[k]}(K):={\\rm vol}_n(K)^{-\\frac{1}{n}}\\left (\\int_{G_{n,k}}{\\rm vol}_k(P_F(K))^{-n}\\,d\\nu_{n,k}(F)\\right )^{-\\frac{1}{kn}}\\leqslant c\\sqrt{n/k},$$ where $c>0$ is an absolute constant. We provide an affirmative answer for some broad classes of random polytopes. We also discuss upper bounds for $\\Phi_{[k]}(K)$ when $K=B_1^n$, the unit ball of $\\ell_1^n$, and explain how this "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.08015","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"}