  The so-called l0 pseudonorm counts the number of nonzero components of a
vector of a Euclidian space. It is well-known that the l0 pseudonorm is not
convex, as its Fenchel biconjugate is zero. In this paper, we introduce a
suitable conjugacy, induced by a novel coupling, E-Capra, that has the property
of being constant along primal rays like the l0 pseudonorm. The coupling
E-Capra belongs to the class of one-sided linear couplings, that we introduce;
we show that they induce conjugacies that share nice properties with the
classic Fenchel conjugacy. For the E-Capra conjugacy, induced by the coupling
E-Capra, we relate the E-Capra conjugate and biconjugate of the l0 pseudonorm,
the characteristic functions of its level sets and the sequence of so-called
top-k norms. In particular, we prove that the l0 pseudonorm is equal to its
biconjugate: hence, the l0 pseudonorm is E-Capra-convex in the sense of
generalized convexity. As a corollary, we show that there exists a proper
convex lower semicontinuous function such that this function and the l0
pseudonorm coincide on the Euclidian unit sphere. This hidden convexity
property is somewhat surprising as the l0 pseudonorm is a highly nonconvex
function of combinatorial nature. We provide different expressions for this
proper convex lower semicontinuous function, and we give explicit formulas in
the two-dimensional case.
