Is there a natural way to order data in dimension greater than one? The approach based on the notion of data depth, often associated with the name of John Tukey, is among the most popular. Tukey’s depth has found applications in robust statistics, the study of elections and social choice, and graph theory. We will give an introduction to the topic (with an emphasis on robust statistics), describe some remaining open questions as well as our recent progress towards the solutions.

This talk is based on the joint work with Yinan Shen.