{"paper":{"title":"Constrained Triangulations, Volumes of Polytopes, and Unit Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Mario Weitzer, Michael Kerber, Robert Tichy","submitted_at":"2016-09-16T12:10:22Z","abstract_excerpt":"Given a polytope $\\mathcal{P}$ in $\\mathbb{R}^d$ and a subset $U$ of its vertices, is there a triangulation of $\\mathcal{P}$ using $d$-simplices that all contain $U$? We answer this question by proving an equivalent and easy-to-check combinatorial criterion for the facets of $\\mathcal{P}$. Our proof relates triangulations of $\\mathcal{P}$ to triangulations of its \"shadow\", a projection to a lower-dimensional space determined by $U$. In particular, we obtain a formula relating the volume of $\\mathcal{P}$ with the volume of its shadow. This leads to an exact formula for the volume of a polytope "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.05017","kind":"arxiv","version":3},"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"}