On the behavior of Tukey's depth and median under symmetric stable distributions

Some curious properties of Tukey's depth and Tukey's multivariate median are revealed by examining their behavior at multivariate distributions possessing independent and identically distributed symmetric stable marginals. In particular, (i) the shape of the contours for Tukey's depth can be the same for large classes of distributions, (ii) the influence function of a linear combination of the components of Tukey's median can be uniformly smaller than the influence function of the univariate median for the corresponding linear combination of the multivariate distribution, and (iii) the maximum bias under epsilon contamination for Tukey's median can be smaller than the maximum bias of the median of some univariate projections of the data.