Leonardo Martínez (Ben-Gurion University)
We introduce the notion of depth with respect to a finite family F of convex sets in R^d that generalizes the well-studied Tukey depth. Specifically, we say that a point p has depth m with respect to F if every hyperplane that contains p intersects at least m sets of F. We study some nice properties of Tukey depth that extend to this definition and point out some key differences.
By imposing additional intersection hypothesis to the family F, we prove a centerpoint theorem for family depth. This result can be thought of as a refinement that interpolates between the classical Rado's centerpoint theorem and Helly's theorem. The main theorem ties centerpoints with a purely combinatorial problem on hitting sets.
Finally, we apply the results and techniques above to geometric transverals theory. We get a new Helly-type theorem for fractional transversal hyperplanes and a new proof for a line transversal theorem of A. Holmsen.