  We consider the minimization of the number of non-zero coefficients (the
$\ell_0$ "norm") of the representation of a data set in terms of a dictionary
under a fidelity constraint. (Both the dictionary and the norm defining the
constraint are arbitrary.) This (nonconvex) optimization problem naturally
leads to the sparsest representations, compared with other functionals instead
of the $\ell_0$ "norm". Our goal is to measure the sets of data yielding a
$K$-sparse solution--i.e. involving $K$ non-zero components. Data are assumed
uniformly distributed on a domain defined by any norm--to be chosen by the
user. A precise description of these sets of data is given and relevant bounds
on the Lebesgue measure of these sets are derived. They naturally lead to bound
the probability of getting a $K$-sparse solution. We also express the
expectation of the number of non-zero components. We further specify these
results in the case of the Euclidean norm, the dictionary being arbitrary.
