  We propose a notion of depth with respect to a finite family $\mathcal{F}$ of
convex sets in $\mathbb{R}^d$ which we call $\text{dep}_\mathcal{F}$. We begin
showing that $\text{dep}_\mathcal{F}$ satisfies some expected properties for a
measure of depth and that this definition is closely related to the notion of
depth proposed by J. Tukey. We show that some properties of Tukey depth extend
to $\text{dep}_\mathcal{F}$ and we point out some key differences.
  We then focus on the following centerpoint-type question: what is the best
depth $\alpha_{d,k}$ that we can guarantee under the hypothesis that the family
$\mathcal{F}$ is $k$-intersecting? We show a key connection between this
problem and a purely combinatorial problem on hitting sets. The relationship is
useful in both directions. On the one hand, for values of $k$ close to $d$ the
combinatorial interpretation gives a good bound for $k$. On the other hand, for
low values of $k$ we can use the classic Rado's centerpoint theorem to get
combinatorial results of independent interest. For intermediate values of $k$
we present a probabilistic framework to improve the bounds and illustrate its
use in the case $k\approx d/2$. These results can be though of as an
interpolation between Helly's theorem and Rado's centerpoint theorem.
  As an application of these results we find a Helly-type theorem for
fractional hyperplane transversals. We also give an alternative and simpler
proof for a transversal result of A. Holmsen.
