  The renormalized volume of hyperbolic manifolds is a quantity motivated by
the AdS/CFT correspondence of string theory and computed via a certain
regularization procedure. The main aim of the present paper is to elucidate its
geometrical meaning. We use another regularization procedure based on surfaces
equidistant to a given convex surface \partial N. The renormalized volume
computed via this procedure is equal to what we call the W-volume of the convex
region N given by the usual volume of N minus the quarter of the integral of
the mean curvature over \partial N. The W-volume satisfies some remarkable
properties. First, this quantity is self-dual in the sense explained in the
paper. Second, it verifies some simple variational formulas analogous to the
classical geometrical Schlafli identities. These variational formulas are
invariant under a certain transformation that replaces the data at \partial N
by those at infinity of M. We use the variational formulas in terms of the data
at infinity to give a simple geometrical proof of results of Takhtajan et al on
the Kahler potential on various moduli spaces.
