  Identification of the center of a data cloud is one of the basic problems in
statistics. One popular choice for such a center is the median, and several
versions of median in finite dimensional spaces have been studied in the
literature. In particular, medians based on different notions of data depth
have been extensively studied by many researchers, who defined median as the
point, where the depth function attains its maximum value. In other words, the
median is the deepest point in the sample space according to that definition.
In this paper, we investigate the deepest point for probability distributions
in infinite dimensional spaces. We show that for some well-known depth
functions like the band depth and the half-region depth in function spaces,
there may not be any meaningful deepest point for many well-known and commonly
used probability models. On the other hand, certain modified versions of those
depth functions as well as the spatial depth function, which can be defined in
any Hilbert space, lead to some useful notions of the deepest point with nice
geometric and statistical properties. The empirical versions of those deepest
points can be conveniently computed for functional data, and we demonstrate
this using some simulated and real data sets.
