VOLUME ESTIMATES FOR ALEXANDROV SPACES WITH CONVEX BOUNDARIES

We estimate the upper bound of volume of a closed positively or nonnegatively curved Alexandrov space X with strictly convex boundary. We also discuss the equality case. In particular, the boundary conjecture holds when the volume upper bound is achieved. Our theorem can also be applied to Riemannian manifolds with nonsmooth boundary, which generalizes Heintze and Karcher's classical volume comparison theorem. Our main tool is the gradient flow of semiconcave functions.