  Whereas the norm of a vector measures amplitude (and is a 1-homogeneous
function), sparsity is measured by the 0-homogeneous l0 pseudonorm, which
counts the number of nonzero components. We propose a family of conjugacies
suitable for the analysis of 0-homogeneous functions. These conjugacies are
derived from couplings between vectors, given by their scalar product divided
by a 1-homogeneous normalizing factor. With this, we characterize the best
convex lower approximation of a 0-homogeneous function on the unit ''ball'' of
a normalization function (i.e. a norm without the requirement of
subadditivity). We do the same with the best convex and 1-homogeneous lower
approximation. In particular, we provide expressions for the tightest convex
lower approximation of the l0 pseudonorm on any unit ball, and we show that the
tightest norm which minorizes the l0 pseudonorm on the unit ball of any lp-norm
is the l1-norm. We also provide the tightest convex lower convex approximation
of the l0 pseudonorm on the unit ball of any norm.
