{"paper":{"title":"Inclusion-exclusion principles for convex hulls and the Euler relation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.PR","authors_text":"Dmitry Zaporozhets, G\\\"unter Last, Zakhar Kabluchko","submitted_at":"2016-03-04T06:15:28Z","abstract_excerpt":"Consider $n$ points $X_1,\\ldots,X_n$ in $\\mathbb R^d$ and denote their convex hull by $\\Pi$. We prove a number of inclusion-exclusion identities for the system of convex hulls $\\Pi_I:=conv(X_i\\colon i\\in I)$, where $I$ ranges over all subsets of $\\{1,\\ldots,n\\}$. For instance, denoting by $c_k(X)$ the number of $k$-element subcollections of $(X_1,\\ldots,X_n)$ whose convex hull contains a point $X\\in\\mathbb R^d$, we prove that $$ c_1(X)-c_2(X)+c_3(X)-\\ldots + (-1)^{n-1} c_n(X) = (-1)^{\\dim \\Pi} $$ for all $X$ in the relative interior of $\\Pi$. This confirms a conjecture of R. Cowan [Adv. Appl. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.01357","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"}