  Recently the authors have introduced a new gauged supergravity theory with a
positive definite potential in D=6, obtained through a generalised Kaluza-Klein
reduction from D=7. Of particular interest is the fact that this theory admits
certain Minkowski x Sphere vacua. In this paper we extend the previous results
by constructing gauged supergravities with positive definitive potentials in
diverse dimensions, together with their vacuum solutions. In addition, we prove
the supersymmetry of the generalised reduction ansatz. We obtain a
supersymmetric solution with no form-field fluxes in the new gauged theory in
D=9. This solution may be lifted to D=10, where it acquires an interpretation
as a time-dependent supersymmetric cosmological solution supported purely by
the dilaton. A further uplift to D=11 yields a solution describing a pp-wave.
