Bishop–Gromov inequality
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In mathematics, the Bishop–Gromov inequality is a classical theorem in Riemannian geometry, named after Richard L. Bishop and Mikhail Gromov. It is the key point in the proof of Gromov's compactness theorem.
[edit] 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
