On angles between convex sets in Hilbert spaces

TL;DR

This paper characterizes the minimal angle between convex cones in Hilbert spaces, providing conditions for positivity, optimal solutions, and the closedness of cone sums, extending classical results to nonlinear convex sets.

Contribution

It generalizes the concept of minimal angles from linear subspaces to convex cones, establishing new conditions for positivity and closedness in Hilbert spaces.

Findings

Necessary conditions for minimal angles between convex cones.

Sufficient conditions for the closedness of the sum of convex cones.

The angles involved cannot both be positive when cones intersect.

Abstract

The notion of the angle between two subspaces has a long history, dating back to Friedrichs's work in 1937 and Dixmier's work on the minimal angle in 1949. In 2006, Deutsch and Hundal studied extensions to convex sets in order to analyze convergence rates for the cyclic projections algorithm. In this work, we characterize the positivity of the minimal angle between two convex cones. We show the existence of, and necessary conditions for, optimal solutions of minimal angle problems associated with two convex subsets as well. Moreover, we generalize a result by Deutsch on minimal angles from linear subspaces to cones. This generalization yields sufficient conditions for the closedness of the sum of two closed convex cones. This also relates to conditions proposed by Beutner and by Seeger and Sossa. Furthermore, we investigate the relation between the intersection of two cones (at least one of which is nonlinear) and the intersection of the polar and dual cones of the underlying cones. It turns out that the two angles involved cannot be positive simultaneously. Various examples illustrate the sharpness of our results.