  Tukey's depth (or halfspace depth) is a widely used measure of centrality for
multivariate data. However, exact computation of Tukey's depth is known to be a
hard problem in high dimensions. As a remedy, randomized approximations of
Tukey's depth have been proposed. In this paper we explore when such randomized
algorithms return a good approximation of Tukey's depth. We study the case when
the data are sampled from a log-concave isotropic distribution. We prove that,
if one requires that the algorithm runs in polynomial time in the dimension,
the randomized algorithm correctly approximates the maximal depth $1/2$ and
depths close to zero. On the other hand, for any point of intermediate depth,
any good approximation requires exponential complexity.
