Tukey's median is among one of the earliest known high breakdown point multivariate location statistics. Aside from its breakdown point, though, little else appears to be known about its robustness properties. In this paper we investigate other aspects of Tukey's median, and in particular we derive and study its influence function and its maximum contamination bias function. When judged by these other robustness criteria, Tukey's median again proves to be highly robust.
"The influence function and maximum bias of Tukey's median." Ann. Statist. 30 (6) 1737 - 1759, December 2002. https://doi.org/10.1214/aos/1043351255