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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 <math>S^m_k<math> a complete simply connected m-dimensional Riemannian manifold of constant sectional curvature <math>k<math>, i.e. an m-sphere of radius <math>1/\sqrt{k}<math> if <math>k>0<math>, Euclidean m-space if <math>k=0<math> and hyperbolic m-space with curvature <math>k<math> if <math>k<0<math>.
Let <math>M<math> be a complete m-dimensional Riemannian manifold with Ricci curvature <math>\ge (m-1)k,<math> <math>p\in M.<math>
Let us denote by <math>v_p(R)<math> the volume of the ball with center p and radius R in <math>M<math> and by <math>V(R)<math> the volume of the ball of radius R in <math>S^m_k.<math>
Then function <math>f_p(R)=v_p(R)/V(R)<math> is nonincreasing for any p.
In particular this implies that for any p and R we have
- <math>v_p(R)\le V(R).<math>
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