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

Abstract

Let $M$ be a complete Riemannian manifold and $\nu$ 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 $\nu$ which ensures existence and uniqueness of the global Riemannian $L^{p}$ center of mass with respect to $\nu$. 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.