About the kernel of the strongly quasiconvex function generated projection
A. B. Németh, S. Z. Németh
TL;DR
The paper generalizes Euclidean projection through strongly quasiconvex functions $\varphi$, embedding this generalized projection into a duality framework of mutually polar cone mappings and vector-lattice theory. It identifies precise conditions under which the projection kernel $\ker(P)$ is itself a closed convex cone, and it characterizes cones and projections that satisfy this property. By examining weighted power-cones, halfspaces, and minimal wedges, the authors derive geometric criteria—such as meridian coherence—for when generalized projections yield mutually polar retractions. They introduce the global bipolar projector concept, linking these geometric and duality properties across all closed convex cones and establishing when a norm is globally bipolar, with SPD-based norms providing concrete realizations.
Abstract
This paper explores a natural generalization of Euclidean projection through the lens of strongly quasiconvex functions, as developed in prior works. By establishing a connection between strongly quasiconvex functions and the theory of mutually polar mappings on convex cones, we integrate this generalized projection concept into the duality framework of Riesz spaces, vector norms, and Euclidean metric projections. A central result of this study is the identification of conditions under which the null space of a projection onto a closed convex cone forms a closed convex cone. We provide a comprehensive characterization of such cones and projections, highlighting their fundamental role in extending the duality theory to generalized projection operators.