The multivariate L-1-median and associated data depth

This paper gives three related results: (i) a new, simple, fast, monotonica lly converging algorithm for deriving the L-1-median of a data cloud in R-d , a problem that can be traced to Fermat and has fascinated applied mathema ticians for over three centuries; (ii) a new general definition for depth f unctions, as functions of multivariate medians, so that different definitio ns of medians will, correspondingly, give rise to different depth functions ; and (iii) a simple closed-form formula of the L-1-depth function for a gi ven data cloud in R-d.