  The so-called l0 pseudonorm, or cardinality function, counts the number of
nonzero components of a vector. In this paper, we analyze the l0 pseudonorm by
means of so-called Capra (constant along primal rays) conjugacies, for which
the underlying source norm and its dual norm are both orthant-strictly
monotonic (a notion that we formally introduce and that encompasses the lp
norms, but for the extreme ones). We obtain three main results. First, we show
that the l0 pseudonorm is equal to its Capra-biconjugate, that is, is a
Capra-convex function. Second, we deduce an unexpected consequence, that we
call convex factorization: the l0 pseudonorm coincides, on the unit sphere of
the source norm, with a proper convex lower semicontinuous function. Third, we
establish a variational formulation for the l0 pseudonorm by means of
generalized top-k dual~norms and k-support dual~norms (that we formally
introduce).
