Combinatorial depth measures are a way to generalize the concept of medians to higher dimensions, formalizing our intuition that some query points lie „deeper“ within a set of data points than others. Several such measures have been introduced in the last century, such as Tukey depth or Simplicial depth. In this talk, I consider families of combinatorial depth measures defined by natural sets of axioms and show that they cannot differ too much.
Many fundamental problems in discrete geometry, such as the centerpoint theorem or Tverberg’s theorem, can be phrased naturally in terms of depth measures. Another example is Gil Kalai’s Cascade conjecture about the dimensions of Tverberg depth regions. This conjecture can be generalized to any depth measure, and in the second part of the talk, I will prove that the conjecture is true for depth measures satisfying the strongest set of axioms, which includes Tukey depth.