Bishop-Gromov inequality Guide, Meaning , Facts, Information and Description
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.
Let us denote by a complete simply connected m-dimensional Riemannian manifold of constant sectional curvature , i.e. an m-sphere of radius if , Euclidean m-space if and hyperbolic m-space with curvature if .
Let be a complete m-dimensional Riemannian manifold with Ricci curvature , .
Let us denote by the volume of the ball with center p and radius R in and by the volume of the ball of radius R in .
Then function is nonincreasing for any p.
In particular this implies that for any p and R we have
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