Jensen's inequality for multivariate medians

Given a probability measure μ on Borel sigma-field of Rd, and a function f:Rd?R, the main issue of this work is to establish inequalities of the type f(m)?M, where m is a median (or a deepest point in the sense explained in the paper) of μ and M is a median (or an appropriate quantile) of the measure μf=μ○f−1. For the most popular choice of halfspace depth, we prove that the Jensen's inequality holds for the class of quasi-convex and lower semi-continuous functions f. To accomplish the task, we give a sequence of results regarding the “type D depth functions” according to classification in Y. Zuo, R. Serfling, General notions of statistical depth function, Ann. Statist. 28 (2000) 461-482], and prove several structural properties of medians, deepest points and depth functions. We introduce a notion of a median with respect to a partial order in Rd and we present a version of Jensen's inequality for such medians. Replacing means in classical Jensen's inequality with medians gives rise to applications in the framework of Pitman's estimation.