  We generalize central-charge relations and differential identities of N=2
Special Geometry to N extended supergravity in any dimension 4 \leq D <10, and
p-extended objects. We study the extremization of the ADM mass per unit of
p-volume of BPS extended objects. Runaway solutions for a ``dilaton'' degree of
freedom leading to a vanishing result are interpreted as BPS extremal states
having vanishing Bekenstein-Hawking Entropy.
