  For multivariate data, Tukey's half-space depth is one of the most popular
depth functions available in the literature. It is conceptually simple and
satisfies several desirable properties of depth functions. The Tukey median,
the multivariate median associated with the half-space depth, is also a
well-known measure of center for multivariate data with several interesting
properties. In this article, we derive and investigate some interesting
properties of half-space depth and its associated multivariate median. These
properties, some of which are counterintuitive, have important statistical
consequences in multivariate analysis. We also investigate a natural extension
of Tukey's half-space depth and the related median for probability
distributions on any Banach space (which may be finite- or
infinite-dimensional) and prove some results that demonstrate anomalous
behavior of half-space depth in infinite-dimensional spaces.
