We study a generalization of the Fréchet 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.