
Journal of Convex Analysis 30 (2023), No. 3, 743769 Copyright Heldermann Verlag 2023 OrthantStrictly Monotonic Norms, Generalized Topk and kSupport Norms and the l_{0} Pseudonorm JeanPhilippe Chancelier CERMICS, Ecole des Ponts, MarnelaVallée, France jeanphilippe.chancelier@enpc.fr Michel De Lara CERMICS, École des Ponts, MarnelaVallée, France, MarnelaVallée, France michel.delara@enpc.fr [Abstractpdf] The socalled $\ell_0$ pseudonorm on the Euclidean space $\mathbb{R}^d$ counts the number of nonzero components of a vector. We say that a sequence of norms is strictly increasingly graded (with respect to the $\ell_0$ pseudonorm) if it is nondecreasing and that the sequence of norms of a vector $x$ becomes stationary exactly at the index $\ell_0(x)$. 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 orthantstrictly monotonic norms (that encompasses the $\ell_p$ norms, but for the extreme ones). Then, we show that an orthantstrictly 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 $\ell_0$ pseudonorm are expressed by means of the difference of two norms. Our results rely on the study of orthantstrictly monotonic norms. Keywords: $\ell_0$ pseudonorm, orthantstrictly monotonic norm, generalized top$k$ norm, generalized $k$support norm, strictly graded sequence of norms. MSC: 15A60, 46N10. [ Fulltextpdf (202 KB)] for subscribers only. 