RIEMANNIAN LpCENTER OF MASS: EXISTENCE, UNIQUENESS, AND CONVEXITY

Let M be a complete Riemannian manifold and ν a probability measure on M. Assume 1 ≤ p ≤ ∞. 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 uniquenes...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2011-02, Vol.139 (2), p.655-673
Main Author: AFSARI, BIJAN
Format: Article
Language:English
Subjects:
Center of mass
Convexity
Cots
Curvature
Geodesy
Hemispheres
Riemann manifold
Uniqueness
Vector fields
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Summary:Let M be a complete Riemannian manifold and ν a probability measure on M. Assume 1 ≤ p ≤ ∞. 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 v. 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.
ISSN:0002-9939
1088-6826