Relatively maximum volume rigidity in alexandrov geometry

Given compact metric spaces X and Z with Hausdorff dimension n, if there is a distance-nonincreasing onto map f : Z → X, then the Hausdorff nvolumes satisfy vol(X)≤ vol(Z) The relatively maximum volume conjecture says that if X and Zare both Alexandrov spaces and vol(X)= vol(Z), X is isometric to a gluing space produced from Z along its boundary ∂ Z and f is length-preserving. We partially verify this conjecture and give a further classification for compact Alexandrovn-spaces with relatively maximum volume in terms of a fixed radius and space of directions. We also give an elementary proof for a pointed version of the Bishop-Gromov relative volume comparison with rigidity in Alexandrov geometry.