Several breakthrough results can be attributed to the project:
1. The PI has achieved the first progress in 25 years towards understanding the true asymptotic growth rates of weak epsilon-nets with respect to convex sets in the Euclidean plane.
The study of epsilon-nets is central to computational geometry, as they determine the performance of the best known algorithms for geometric hitting set/set-cover problems, and are used in most divide-and-conquer schemes.
It is a decades-old open problem to determine whether the growth rates of (optimal) weak-epsilon nets with respect to convex sets in the Euclidean space overly resemble those of strong epsilon-nets.
We achieve the first progress in 25 years by improving the 1992 planar bound of Alon, Barany, Furedi and Kleitman.
A series of 2 papers establishes improved bounds for weak epsilon-nets in all dimensions. It is based upon a curious reduction of the problem to incidence bounds between points and generalized combinatorial hyperplanes.
2. The PI's collaboration with J. Pach and G. Tardos shows that any finite set of points in the Euclidean plane determines almost linearly many pairwise crossing segments, and the result extends to all sufficiently dense geometric graphs.
Notice that the cornerstone Crossing Lemma (1982) in combinatorial geometry yields a tight relation between the number of edges in a geometric graph and the number of edge crossings in its planar embedding. The result bears enormous importance to computational geometry.
Unfortunately, the Crossing Lemma tells very little, if at all, of the underlying intersection structure (i.e. intersection graph) of the edges, and very little progress has been achieved in the passed decades.
Our lower bound on pairwise crossing edges is almost tight, and improves a much weaker 1990 lower bound of Aronov, Erdos, Goddard, Klugerman, Kleitman, Pach and Schulman.
3. Another collaboration of the PI with J. Pach and G. Tardos establishes an analogue of the Crossing Lemma for contact graphs of Jordan curves in the plane. That is, we establish a non-trivial relation between the number of touchings (i.e. points of tangency which occur if the two involved curves intersect only once) and the overall number of their intersection points in a family of Jordan curves. In particular, we establish the 1995 Richter-Thomassen Conjecture for families of closed Jordan curves.
In addition, a joint work of the PI with the project postdoc Leonardo Martínez-Sandoval, and the team visitor Edgardo Roldán-Pensado resulted in a a novel Colourful Helly Theorem which offers a complementary relation between transversals of various dimensions to a family of convex sets that satisfies Lovasz's Colourful Helly Hypothesis. As a by-product, this study introduced to the discrete geometry community an intriguing conjecture on lines that cross a family of pairwise intersecting convex sets in the Euclidean 3-space.
