  The support of a vector in R d is the set of indices with nonzero entries.
Functions of the support have the property to be 0-homogeneous and, because of
that, the Fenchel conjugacy fails to provide relevant analysis. In this paper,
we define the coupling Capra between R d and itself by dividing the classic
Fenchel scalar product coupling by a given (source) norm on R d. Our main
result is that, when both the source norm and its dual norm are
orthant-strictly monotonic, any nondecreasing finite-valued function of the
support mapping is Capra-convex, that is, is equal to its Capra-biconjugate
(generalized convexity). We also establish that any such function is the
composition of a proper convex lower semi continuous function on R d with the
normalization mapping on the unit sphere (hidden convexity), and that, when
normalized, it admits a variational formulation, which involves a family of
generalized local-K-support dual norms.
