  In this note, we estimate the upper bound of volume of 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 also can be applied to
Riemannian manifolds with non-smooth boundary, which generalizes Heintze and
Karcher's classical volume comparison theorem. Our main tool is the gradient
flow of semi-concave functions.
