Riemannian 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 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.

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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

Email:
bijan@umd.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-2010-10541-5

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.