Bishop–Gromov inequality
In mathematics, the Bishop–Gromov inequality is a comparison theorem in Riemannian geometry, named after Richard L. Bishop and Mikhail Gromov. It is closely related to Myers' theorem, and is the key point in the proof of Gromov's compactness theorem. Statement. Let formula_1 be a complete "n"-dimensional Riemannian manifold whose Ricci curvature satisfies the lower bound for a constant formula_3. Let formula_4 be the complete "n"-dimensional simply connected space of constant sectional curvature formula_5 (and hence of constant Ricci curvature formula_6); thus formula_4 is the "n"-sphere of radius formula_8 if formula_9, or "n"-dimensional Euclidean space if  formula_10, or an appropriately rescaled version of "n"-dimensional hyperbolic space if formula_11. Denote by formula_12 the ball of radius "r" around a point "p", defined with respect to the Riemannian distance function. Then, for any formula_13 and formula_14, the function is non-increasing on formula_16. As "r" goes to zero, the ratio approaches one, so together with the monotonicity this implies that This is the version first proved by Bishop.