It is shown how dimensional reduction of a (4 + N)-dimensional theory can lead to an effective four-dimensional, broken-symmetry theory of gravity, whose lagrangian density is of the form L = 1 2 ϵπ 2R + 1 2 π ;kπ ;k − V(π) , where R is the Ricci scalar, the field π is the inverse of the radius function of the internal space, and the potential contains both a classical contribution and a Casimir energy − 1 2 λ′π 4 . For small π, the Casimir energy dominates, and quantum corrections automatically generate a Coleman-Weinberg potential V(π) = 1 4 λπ 4[ ln( π 2 μ 2 ) − 1 2 ] + λ A generalized Rubakov-Shaposhnikov ansatz is made in the (4 + N)-dimensional metric, by means of which the vacuum energy density can be made sufficiently small for compatibility with the cosmological microwave background radiation, for a given λ. It turns out that ϵ = 2( N 2 + 2 N − 12) −1. Cosmological inflation can be realized, provided that N ⪆ 20.