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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.

[edit] 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>