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.

两个子空间之间的角度概念由来已久，其历史可追溯至1937年的Friedrichs和1949年的Dixmier关于最小角度的工作。2006年，Deutsch和Hundal研究了凸集的扩展，以便分析循环的收敛速度。投影算法。在这项工作中，我们刻画了两个凸锥之间的最小角度的正性。我们还显示了与两个凸子集相关的最小角度问题的最优解的存在性和必要条件。此外，我们将Deutsch的结果推广到从线性子空间到圆锥的最小角度。这种概括为两个封闭的凸锥之和的封闭性提供了充分的条件。这也与Beutner以及Seeger和Sossa提出的条件有关。此外，我们研究了两个圆锥（至少其中一个是非线性的）的相交与基础圆锥的极圆锥和双圆锥的相交之间的关系。事实证明，所涉及的两个角度不能同时为正。各种各样的例子说明了我们结果的鲜明性。