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Bishop-Gromov inequality
In mathematics, the Bishop-Gromov inequality is a classical theorem in Riemannian geometry. It is the key point in the proof of Gromov's compactness theorem.
Statement
Let us denote by 
a complete simply connected m-dimensional Riemannian manifold of constant sectional curvature k, i.e. an m-sphere of radius 
if k > 0, Euclidean m-space if k = 0 and hyperbolic m-space with curvature k if k < 0.
Let M be a complete m-dimensional Riemannian manifold with Ricci curvature 
Let us denote by vp(R) the volume of the ball with center p and radius R in M and by V(R) the volume of the ball of radius R in 
Then function fp(R) = vp(R) / V(R) is nonincreasing for any p.
In particular this implies that for any p and R we have
10-26-2009 08:16:03
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The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details