
Open access
Date
2019Type
- Conference Paper
Abstract
The Tverberg theorem is one of the cornerstones of discrete geometry. It states that, given a set X of at least (d+1)(r-1)+1 points in R^d, one can find a partition X=X_1 cup ... cup X_r of X, such that the convex hulls of the X_i, i=1,...,r, all share a common point. In this paper, we prove a strengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed. As a concrete application, we show that any n points in the plane in general position span floor[n/3] vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Alvarez-Rebollar et al. guarantees floor[n/6] pairwise crossing triangles. Our result generalizes to a result about simplices in R^d,d >=2. Show more
Permanent link
https://doi.org/10.3929/ethz-b-000351624Publication status
publishedExternal links
Book title
35th International Symposium on Computational Geometry (SoCG 2019)Journal / series
Leibniz International Proceedings in Informatics (LIPIcs)Volume
Pages / Article No.
Publisher
Schloss Dagstuhl – Leibniz-Zentrum für InformatikEvent
Subject
Discrete geometry; Tverberg theorem; Crossing Tverberg theoremOrganisational unit
03457 - Welzl, Emo (emeritus) / Welzl, Emo (emeritus)
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