Volume estimates for Alexandrov spaces with convex boundaries

In this note, we estimate the upper bound of volume closed positively or nonnegatively curved Alexandrov space $X$ with strictly convex boundary. We also discuss equality case. particular, Boundary Conjecture holds when is achieved. Our theorem can be applied to Riemannian manifolds non-smooth boundary, which generalizes Heintze and Karcher's classical comparison theorem. main tool gradient flow semi-concave functions.