# Depth with respect to a family of convex sets

### Wednesday, March 15th, 2017 16:10

### Schreiber 309

### Depth with respect to a family of convex sets

**Leonardo Martinez, Ben-Gurion University**

### Abstract:

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 and hitting sets in a purely combinatorial problem.

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.