We describe multivariate generalizations of the median, trimmed mean and $W$ estimates. The estimates are based on a geometric construction related to "projection pursuit." They are both affine equivariant (coordinate-free) and have high breakdown point. The generalization of the median has a breakdown point of at least $1/(d + 1)$ in dimension $d$ and the breakdown point can be as high as $1/3$ under symmetry. In contrast, various estimators based on rejecting apparent outliers and taking the mean of the remaining observations have breakdown points not larger than $1/(d + 1)$ in dimension $d$.
"Breakdown Properties of Location Estimates Based on Halfspace Depth and Projected Outlyingness." Ann. Statist. 20 (4) 1803 - 1827, December, 1992. https://doi.org/10.1214/aos/1176348890