2026-07-21T14:13:43Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:19997 2024-06-06T06:21:15Z ddc:004 open_access

Colorful Intersections and Tverberg Partitions Dobbins, Michael Gene Holmsen, Andreas F. Lee, Dohyeon Tverberg’s theorem geometric transversals topological combinatorics configuration space/test map discrete Morse theory The colorful Helly theorem and Tverberg’s theorem are fundamental results in discrete geometry. We prove a theorem which interpolates between the two. In particular, we show the following for any integers d ≥ m ≥ 1 and k a prime power. Suppose F₁, F₂, … , F_m are families of convex sets in ℝ^d, each of size n > (d/m+1)(k-1), such that for any choice C_i ∈ F_i we have ⋂_{i = 1}^m C_i ≠ ∅. Then, one of the families F_i admits a Tverberg k-partition. That is, one of the F_i can be partitioned into k nonempty parts such that the convex hulls of the parts have nonempty intersection. As a corollary, we also obtain a result concerning r-dimensional transversals to families of convex sets in ℝ^d that satisfy the colorful Helly hypothesis, which extends the work of Karasev and Montejano. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Michael Gene Dobbins and Andreas F. Holmsen and Dohyeon Lee 2024 Is Part Of LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2024.52 urn:nbn:de:0030-drops-199973 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.52 eng https://creativecommons.org/licenses/by/4.0/legalcode