  For any given partial order in a $d$-dimensional euclidean space, under mild
regularity assumptions, we show that the intersection of closed (generalized)
intervals containing more than 1/2 of the probability mass, is a non-empty
compact interval. This property is shared with common intervals on real line,
where the intersection is the median set of the underlying probability
distribution. So obtained multivariate medians with respect to a partial order,
can be observed as special cases of centers of distribution in the sense of
type D depth functions introduced by Y. Zuo and R. Serfling, {\em Ann. Stat.},
{\bf 28} (2000), 461-482. We show that the halfspace depth function can be
realized via compact convex sets, or, for example, closed balls, in place of
halfspaces, and discuss structural properties of halfspace and related depth
functions and their centers. Among other things, we prove that, in general, the
maximal guaranteed depth is $\frac{1}{d+1}$. As an application of these
results, we provide a Jensen's type inequality for functions of several
variables, with medians in place of expectations, which is an extension of the
previous work by M. Merkle, {\em Stat. Prob. Letters}, {\bf 71} (2005),
277--281.}
