Approximate selections in $\alpha$-convex metric spaces and topological degree

Abstract

The existence of continuous approximate selections is proved for a class of upper semicontinuous multifunctions taking closed $\alpha$-convex values in a metric space equipped with an appropriate notion of $\alpha$-convexity. The approach is based on the definition of pseudo-barycenter of an ordered $n$-tuple of points. As an application, a notion of topological degree for a class of $\alpha$-convex multifunctions is developed.