Consider a nonempty finite set of nonzero vectors \(S \subset \mathbb {R}^n\). The angle between a nonzero vector \(v \in \mathbb {R}^n\) and S is the smallest angle between v and an element of S. The cosine measure of S is the cosine of the largest possible angle between a nonzero vector \(v \in \mathbb {R}^n\) and S. The cosine measure provides a way of quantifying the positive spanning property of a set of vectors, which is important in the area of derivative-free optimization. This paper proves some of the properties of the cosine measure for a nonempty finite set of nonzero vectors. It also introduces the notion of the uniform angle subspace and some cones associated with it and proves some of their properties. Moreover, this paper proves some results that characterize the Karush–Kuhn–Tucker (KKT) points for the optimization problem of calculating the cosine measure. These characterizations of the KKT points involve the uniform angle subspace and its associated cones. Finally, this paper provides an outline for calculating the cosine measure of any nonempty finite set of nonzero vectors.

考虑非零向量\（S \ subset \ mathbb {R} ^ n \）的非空有限集。非零向量\（v \ in \ mathbb {R} ^ n \）与S之间的角度是v与S的元素之间的最小角度。S的余弦量度是非零向量\（v \ in \ mathbb {R} ^ n \）与S之间的最大可能角度的余弦。余弦测度提供了一种量化一组向量的正跨度性质的方法，这在无导数优化领域很重要。本文证明了非零向量的非空有限集的余弦测度的某些性质。它还介绍了均匀角度子空间和与之相关的一些圆锥的概念，并证明了它们的一些特性。此外，本文证明了一些结果，这些结果可表征Karush–Kuhn–Tucker（KKT）点，以解决余弦量度的优化问题。KKT点的这些特征涉及均匀角度子空间及其关联的圆锥。最后，本文提供了一个概述，用于计算非零向量的任何非空有限集的余弦度量。