A Tverberg-type result on multicolored simplices

AbstractLet P1, P2,…, Pd+1 be pairwise disjoint n-element point sets in general position in d-space. It is shown that there exist a point O and suitable subsets Qi ⊆ Pi (i = 1, 2,…, d + 1) such that Qi ≥ cdPi, an every d-dimensional simplex with exactly one vertex in each Qi contains Q in its interior. Here cd is a positive constant depending only on d

Point selections and weak e-nets for convex hulls

One of our results: let X be a finite set on the plane, 0 < g < 1, then there exists a set F (a weak g-net) of size at most 7/e 2 such that every convex set containing at least e\X\ elements of X intersects F. Note that the size of F is independent of the size of X. 1