Some intriguing properties of Tukey’s half-space depth

Tukey’s half-space depth is one of the most popular depth functions available in the literature. It is conceptually very simple, satisfies several desirable properties of depth functions, and can have a natural extension for probability distributions on any Banach space, which may be finite or infinite dimensional. In this article, we derive some counter intuitive properties of half-space depth for probability distributions in finite and infinite dimensional spaces