  Given a set of vectors $\F=\{f_1,\dots,f_m\}$ in a Hilbert space $\HH$, and
given a family $\CC$ of closed subspaces of $\HH$, the {\it subspace clustering
problem} consists in finding a union of subspaces in $\CC$ that best
approximates (models) the data $\F$.
  This problem has applications and connections to many areas of mathematics,
computer science and engineering such as the Generalized Principle Component
Analysis (GPCA), learning theory, compressed sensing, and sampling with finite
rate of innovation. In this paper, we characterize families of subspaces $\CC$
for which such a best approximation exists. In finite dimensions the
characterization is in terms of the convex hull of an augmented set $\CC^+$. In
infinite dimensions however, the characterization is in terms of a new but
related notion of contact hull. As an application, the existence of best
approximations from $\pi(G)$-invariant families $\CC$ of unitary
representations of abelian groups is derived.
