Some eigenvector theorems proved by a Fan-KKM theorem (Q5955063)

Let \(C\) be a closed convex subset of a normed linear space \(X\) and \(f:C\to X\) a continuous map satisfying \(0\not\in f(c)\). The author gives conditions under which there exists \(x_0\in C\) which is the point of best approximation to the ray \(\{tf(x_0): t\in\mathbb{R}\}\) in \(C\).