Resumen de Riemannian Lp center of mass: existence, uniqueness, and convexity

Bijan Afsari

• Let be a complete Riemannian manifold and a probability measure on . Assume . We derive a new bound (in terms of , the injectivity radius of and an upper bound on the sectional curvatures of ) on the radius of a ball containing the support of which ensures existence and uniqueness of the global Riemannian center of mass with respect to . A significant consequence of our result is that under the best available existence and uniqueness conditions for the so-called ``local'' 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 , under the existence and uniqueness conditions, the (global) center of mass belongs to the closure of the convex hull of the masses. We also give a refined result when is of constant curvature.