Multivariate L1 mean

The center of a univariate data set {x1,…,xn} can be defined as the point μ that minimizes the norm of the vector of distances y′=(|x1−μ|,…,|xn−μ|). As the median and the mean are the minimizers of respectively the L1- and the L2-norm of y, they are two alternatives to describe the center of a univariate data set. The center μ of a multivariate data set {x1,…,xn} can also be defined as minimizer of the norm of a vector of distances. In multivariate situations however, there are several kinds of distances. In this note, we consider the vector of L1-distances y′1=(∥x1- μ∥1,…,∥xn- μ∥1) and the vector of L2-distances y′2=(∥x1- μ∥2,…,∥xn-μ∥2). We define the L1-median and the L1-mean as the minimizers of respectively the L1- and the L2-norm of y1; and then the L2-median and the L2-mean as the minimizers of respectively the L1- and the L2-norm of y2. In doing so, we obtain four alternatives to describe the center of a multivariate data set. While three of them have been already investigated in the statistical literature, the L1-mean appears to be a new concept.