Riemannian $L^{p}$ center of mass: Existence, uniqueness, and convexity

Abstract

Let $ M$ be a complete Riemannian manifold and $ ν$ a probability measure on $ M$. Assume $ 1\leq p\leq \infty$. We derive a new bound (in terms of $ p$, the injectivity radius of $ M$ and an upper bound on the sectional curvatures of $ M$) on the radius of a ball containing the support of $ ν$ which ensures existence and uniqueness of the global Riemannian $ L^{p}$ center of mass with respect to $ ν$. A significant consequence of our result is that under the best available existence and uniqueness conditions for the so-called ``local'' $ L^{p}$ center of mass, the global and local centers coincide. In our derivation we also give an alternative proof for a uniqueness result by W. S. Kendall. As another contribution, we show that for a discrete probability measure on $ M$, under the existence and uniqueness conditions, the (global) $ L^{p}$ center of mass belongs to the closure of the convex hull of the masses. We also give a refined result when $ M$ is of constant curvature.