On angles between convex cones

TL;DR

This paper extends classical results on angles between linear subspaces to convex cones, providing new inequalities and examples, and discusses the technical challenges involved in these generalizations.

Contribution

It introduces novel extensions of angle results from subspaces to convex cones, including a conical version of Hundal's inequality and related technical insights.

Findings

Hundal's inequality has a conical extension.

Extensions to Krein, Krasnoselskii, Milman, and Solmon's results are more complex.

Examples demonstrate the sharpness of the new results.

Abstract

There are two basic angles associated with a pair of linear subspaces: the Diximier angle and the Friedrichs angle. The Dixmier angle of the pair of orthogonal complements is the same as the Dixmier angle of the original pair provided that the original pair gives rise to a direct (not necessarily orthogonal) sum of the underlying Hilbert space. The Friedrichs angles of the original pair and the pair of the orthogonal complements always coincide. These two results are due to Krein, Krasnoselskii, and Milman and to Solmon, respectively. In 1995, Deutsch provided a very nice survey with complete proofs and interesting historical comments. One key result in Deutsch's survey was an inequality for Dixmier angles provided by Hundal. In this paper, we present extensions of these results to the case when the linear subspaces are only required to be convex cones. It turns out that Hundal's result has a nice conical extension while the situation is more technical for the results by Krein et al.\ and by Solmon. Our analysis is based on Deutsch's survey and our recent work on angles between convex sets. Throughout, we also provide examples illustrating the sharpness of our results.