  Tukey's halfspace depth can be seen as a stochastic program and as such it is
not guarded against optimizer's curse, so that a limited training sample may
easily result in a poor out-of-sample performance. We propose a generalized
halfspace depth concept relying on the recent advances in distributionally
robust optimization, where every halfspace is examined using the respective
worst-case distribution in the Wasserstein ball of radius $\delta\geq 0$
centered at the empirical law. This new depth can be seen as a smoothed and
regularized classical halfspace depth which is retrieved as $\delta\downarrow
0$. It inherits most of the main properties of the latter and, additionally,
enjoys various new attractive features such as continuity and strict positivity
beyond the convex hull of the support. We provide numerical illustrations of
the new depth and its advantages, and develop some fundamental theory. In
particular, we study the upper level sets and the median region including their
breakdown properties.
