{"paper":{"title":"Monotonicity of the Sample Range of 3-D Data: Moments of Volumes of Random Tetrahedra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Benjamin Reichenwallner, Matthias Reitzner, Stefan Kunis","submitted_at":"2016-12-06T16:28:15Z","abstract_excerpt":"The sample range of uniform random points $X_1, \\dots , X_n$ chosen in a given convex set is the convex hull ${\\rm conv}[X_1, \\dots, X_n]$. It is shown that in dimension three the expected volume of the sample range is not monotone with respect to set inclusion. This answers a question by Meckes in the negative.\n  The given counterexample is the three-dimensional tetrahedron together with an infinitesimal variation of it. As side result we obtain an explicit formula for all even moments of the volume of a random simplex which is the convex hull of three uniform random points in the tetrahedron"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.01893","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"}