We show that the existence of a maximal embedded tube in a hyperbolic n-manifold implies the existence of a certain conical region. One application is to establish a lower bound on the volume of the region outside the tube, thereby improving estimates on volume and estimates on lengths of geodesics in small volume hyperbolic 3-manifolds. We also provide new bounds on the injectivity radius and diameter of an n-manifold.
"Cones Embedded in Hyperbolic Manifolds." J. Differential Geom. 58 (2) 219 - 232, June, 2001. https://doi.org/10.4310/jdg/1090348325