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. However, the mathematical expression of the l0 pseudonorm, taking integer values, makes it difficult to handle in optimization problems on Rd. Moreover, the Fenchel conjugacy fails to provide relevant insight. In this paper, we analyze the l0 pseudonorm by means of a family of so-called Capra conjugacies. For this purpose, to each (source) norm on Rd, we attach a so-called Capra coupling between Rd and Rd, and sequences of generalized top-k dual norms and k-support dual norms. When we suppose that both the source norm and its dual norm are orthant-strictly monotonic, we obtain three main results. First, we show that the l0 pseudonorm is Capra biconjugate, that is, a Capra-convex function. Second, we deduce an unexpected consequence: the l0 pseudonorm coincides, on the unit sphere of the source norm, with a proper convex convex lower semicontinuous (lsc) function on Rd. Third, we establish variational formulations for the l0 pseudonorm and, with these novel expressions, we provide, on the one hand, a new family of lower and upper bounds for the l0 pseudonorm, as a ratio between two norms, and, on the other hand, reformulations for exact sparse optimization problems
