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 .