Uniform Consistency of Generalized Fr\'echet Means

We study a generalization of the Fr\'echet mean on metric spaces, which we call φ-means. Our generalization is indexed by a convex function φ. We find necessary and sufficient conditions for φ-means to be finite and provide a tight bound for the diameter of the intrinsic mean set. We also provide sufficient conditions under which all the φ-means coincide in a single point. Then, we prove the consistency of the sample φ-mean to its population analogue. We also find conditions under which classes of φ-means converge uniformly, providing a Glivenko-Cantelli result. Finally, we illustrate applications of our results and provide algorithms for the computation of φ-means.