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

Author: Bijan Afsari
Journal: Proc. Amer. Math. Soc. 139 (2011), 655-673
MSC (2010): Primary 53C20; Secondary 62H11, 92C55
Published electronically: August 27, 2010
MathSciNet review: 2736346
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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.

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Additional Information

Bijan Afsari
Affiliation: Department of Applied Mathematics, University of Maryland, College Park, Maryland 20742
Address at time of publication: Institute for Systems Research, University of Maryland, College Park, Maryland 20742

Keywords: Riemannian center of mass, Fréchet mean, barycenter, centroid, convex hull, manifold-valued data, comparison theorems
Received by editor(s): July 14, 2009
Received by editor(s) in revised form: July 15, 2009, February 25, 2010, March 1, 2010, and April 12, 2010
Published electronically: August 27, 2010
Additional Notes: This research was supported in part by the Army Research Office under the ODDR&E MURI01 Program Grant No. DAAD19-01-1-0465 to the Center for Communicating Networked Control Systems (through Boston University), by NSF-NIH Collaborative Research in Computational Neuroscience Program (CRCNS2004), NIH-NIBIB grant 1 R01 EB004750-1, and by NSF grants DMS-0204671 and DMS-0706791.
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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