  We analyze the performance of the Tukey median estimator under total
variation (TV) distance corruptions. Previous results show that under Huber's
additive corruption model, the breakdown point is 1/3 for high-dimensional
halfspace-symmetric distributions. We show that under TV corruptions, the
breakdown point reduces to 1/4 for the same set of distributions. We also show
that a certain projection algorithm can attain the optimal breakdown point of
1/2. Both the Tukey median estimator and the projection algorithm achieve
sample complexity linear in dimension.
