This paper gives three related results: ( i ) a new, simple, fast, monotonically converging algorithm for deriving the L1 -median of a data cloud in ℝ d , a problem that can be traced to Fermat and has fascinated applied mathematicians for over three centuries; ( ii ) a new general definition for depth functions, as functions of multivariate medians, so that different definitions of medians will, correspondingly, give rise to different dept functions; and ( iii ) a simple closed-form formula of the L1 -depth function for a given data cloud in ℝ d .