In this paper, we first introduce convexity index \(\alpha \in [0,1]\) for star-shaped sets so that a closed star-shaped set of a Banach space is convex if and only if \(\alpha =1\) . Then we extend Schauder’s fixed point theorem in the following manner (which is even new for compact convex sets): Suppose that X is a Banach space. If S is a compact star-shaped subset with respect to \(p\in S\) with convexity index \(\alpha _p>0\) , then every continuous self-mapping \(f{:}\,S\rightarrow S\) has one of the following two properties: (a) The point p is a fixed point of f, i.e., \(f(p)=p\) ;
(b) f has uncountably many different eigenvalues and eigenvectors; that is, there exists an injective mapping \(\lambda \rightarrow x_\lambda \) from (0, 1] into S such that \( f(x_\lambda )=p+\frac{1}{\lambda \alpha _p}(x_\lambda -p). \)
