Bounded Geometry

Bounded geometry for a uniformly discrete metric space means that balls of any fixed radius contain a uniformly bounded number of points. Precisely, for every finite ,

Uniform discreteness means that distances between distinct points have a positive lower bound. For graph path metrics, a common finite bound on vertex degree ensures the required bounds on balls.

This convention differs from differential-geometric uses of bounded geometry that impose curvature and injectivity-radius conditions on manifolds. Bounded geometry is a hypothesis in the coarse Novikov conjecture.