The projection onto the cross

TL;DR

This paper investigates the geometric and analytical properties of the set of orthogonal vector pairs in Hilbert spaces, revealing their non-closedness in infinite dimensions but establishing their proximinality with explicit projection formulas.

Contribution

It provides the first comprehensive analysis of the projection onto the cross in infinite-dimensional Hilbert spaces, including explicit formulas and closure properties.

Findings

Crosses are never weakly closed in infinite-dimensional spaces.

Crosses are proximinal and admit explicit projections.

The work extends understanding of nonconvex sets in Hilbert spaces.

Abstract

We consider the set of pairs of orthogonal vectors in Hilbert space, which is also called the cross because it is the union of the horizontal and vertical axes in the Euclidean plane when the underlying space is the real line. Crosses, which are nonconvex sets, play a significant role in various branches of nonsmooth analysis such as feasibility problems and optimization problems. In this work, we study crosses and show that in infinite-dimensional settings, they are never weakly (sequentially) closed. Nonetheless, crosses do turn out to be proximinal (i.e., they always admit projections) and we provide explicit formulas for the projection onto the cross in all cases.