  The inscribed radius of a compact manifold with boundary is bounded above if
its Ricci curvature and mean curvature are bounded from below. The rigidity
result implies that the upper bound can be achieved only in space form. In this
paper, we generalize this result to asymptotically hyperbolic Einstein
manifold. We get an upper bound of the relative volume of AH manifold and if we
combine it with the recent work of Wang and Zhou, then the rigidity is
obtained.

Differential Geometry