  The Tukey depth of a flat with respect to a point set is a concept that
appears in many areas of discrete and computational geometry. In particular,
the study of centerpoints, center transversals, Ham Sandwich cuts, or $k$-edges
can all be phrased in terms of depths of certain flats with respect to one or
more point sets. In this work, we introduce the Tukey depth histogram of
$k$-flats in $\mathbb{R}^d$ with respect to a point set $P$, which is a vector
$D^{k,d}(P)$, whose $i$'th entry $D^{k,d}_i(P)$ denotes the number of $k$-flats
spanned by $k+1$ points of $P$ that have Tukey depth $i$ with respect to $P$.
As our main result, we give a complete characterization of the depth histograms
of points, that is, for any dimension $d$ we give a description of all possible
histograms $D^{0,d}(P)$. This then allows us to compute the exact number of
possible such histograms.
