Some New Results on Geometric Transversals

Abstract

We investigate a number of questions, problems, and conjectures related to geometric transversal theory. Among our results we disprove a conjecture of Barany and Kalai regarding weak epsilon-nets for k-flats and convex sets in R-d, and we prove a conjecture of Arocha, Bracho, and Montejano regarding a colorful version of the Goodman-Pollack-Wenger transversal theorem. We also investigate the connected components of the space of line transversals to pairwise disjoint convex sets in R-3, and we extend a theorem of Karasev and Montejano regarding colorful intersections and k-transversals.