Sparse optimization seeks an optimal solution with few nonzero entries. To
achieve this, it is common to add to the criterion a penalty term proportional
to the $\ell_1$-norm, which is recognized as the archetype of sparsity-inducing
norms. In this approach, the number of nonzero entries is not controlled a
priori. By contrast, in this paper, we focus on finding an optimal solution
with at most~$k$ nonzero coordinates (or for short, $k$-sparse vectors), where
$k$ is a given sparsity level (or ``sparsity budget''). For this purpose, we
study the class of generalized $k$-support norms that arise from a given source
norm. When added as a penalty term, we provide conditions under which such
generalized $k$-support norms promote $k$-sparse solutions. The result follows
from an analysis of the exposed faces of closed convex sets generated by
$k$-sparse vectors, and of how primal support identification can be deduced
from dual information. Finally, we study some of the geometric properties of
the unit balls for the $k$-support norms and their dual norms when the source
norm belongs to the family of $\ell_p$-norms.