  Let Z be an Alexandrov space with curvature bounded below by -1 such that Z
is homotopy equivalent to a real hyperbolic manifold M. It is known that the
volume of Z is not smaller than the volume of M. If the volumes are equal, this
short paper proves that the homotopy equivalence is homotopic to an isometric
homeomorphism. The main analytic tool is a theorem of Reshetnyak about
quasiregular maps.
