论文标题
在希尔伯特空间中的凸组之间的角度
On angles between convex sets in Hilbert spaces
论文作者
论文摘要
两个子空间之间的角度的概念具有悠久的历史，可以追溯到弗里德里奇（Friedrichs）在1937年的作品，以及迪克米尔（Dixmier）在1949年的最小角度的工作。2006年，德意志和洪达尔（Deutsch）和洪达尔（Hundal）的扩展是为了分析循环投射率的cosevens sets sets set set。 在这项工作中，我们表征了两个凸锥之间最小角的阳性。我们还显示了与两个凸子集相关的最小角度问题的最佳解决方案的存在和必要条件。此外，我们通过Deutsch从线性子空间到锥的最小角度概括了结果。该概括为两个闭合凸锥的封闭性提供了足够的条件。这也与Beutner以及Seeger和Sossa提出的条件有关。此外，我们研究了两个锥体的交点（至少一个是非线性）与下层锥的极性和双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是积极的。各种例子说明了我们结果的清晰度。
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.
