  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.
