  The so-called l0 pseudonorm on Rd counts the number of nonzero components of
a vector. It is used in sparse optimization, either as criterion or in the
constraints, to obtain solutions with few nonzero entries. For such problems,
the Fenchel conjugacy fails to provide relevant analysis: indeed, the Fenchel
conjugate of the characteristic function of the level sets of the l0 pseudonorm
is minus infinity, and the Fenchel biconjugate of the l0 pseudonorm is zero. In
this paper, we display a class of conjugacies that are suitable for the l0
pseudonorm. For this purpose, we suppose given a (source) norm on Rd. With this
norm, we define, on the one hand, a sequence of so-called coordinate-k norms
and, on the other hand, a coupling between Rd and Rd , called Capra (constant
along primal rays). Then, we provide formulas for the Capra-conjugate and
biconjugate, and for the Capra subdifferentials, of functions of the l0
pseudonorm (hence, in particular, of the l0 pseudonorm itself and of the
characteristic functions of its level sets), in terms of the coordinate-k
norms. As an application, we provide a new family of lower bounds for the l0
pseudonorm, as a fraction between two norms, the denominator being any norm.
