  The so-called l0 pseudonorm on the Euclidean space Rd counts the number of
nonzero components of a vector. We say that a sequence of norms is strictly
increasingly graded (with respect to the l0 pseudonorm) if it is nondecreasing
and that the sequence of norms of a vector~x becomes stationary exactly at the
index l0(x). In this paper, with any (source) norm, we associate sequences of
generalized top-k and k-support norms, and we also introduce the new class of
orthant-strictly monotonic norms (that encompasses the lp norms, but for the
extreme ones). Then, we show that an orthant-strictly monotonic source norm
generates a sequence of generalized top-k norms which is strictly increasingly
graded. With this, we provide a systematic way to generate sequences of norms
with which the level sets of the l0 pseudonorm are expressed by means of the
difference of two norms. Our results rely on the study of orthant-strictly
monotonic norms.
