On the Tukey depth of an atomic measure

This paper gives a relation between the convex Tukey trimmed region (see [J.C. Massé, R. Theodorescu, Halfplane trimming for bivariate distributions, J. Multivariate Anal. 48(2) (1994) 188–202]) of an atomic measure and the support of the measure. It is shown that an atomic measure is concentrated on the extreme points of its Tukey trimmed region. A property concerning the extreme points which have 0 mass is given. As a corollary, we give a new method of proof of the Koshevoy characterization result.