The Crossing Tverberg Theorem

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 ∪ ⋯ ∪ 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 ⌊ 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 Álvarez-Rebollar et al. guarantees ⌊ n / 6 ⌋ pairwise crossing triangles. Our result generalizes to a result about simplices in R d , d ≥ 2 .