[en] 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)epsilonphi²R+(1/2)phisub(;k)phisup(;k)-V(phi), where R is the Ricci scalar, the field phi is the inverse of the radius function of the internal space, and the potential contains both a classical contribution and a Casimir energy -(1/4)lambdaprimephi⁴. For small phi, the Casimir energy dominates, and quantum corrections automatically generate a Coleman-Weinberg potential V(phi)=(1/4)lambdaphi⁴[ln(phi²/μ²)-(1/2)]+Λ-tilde. 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 lambda. It turns out that epsilon=(2/3)(N+1)^-1(N-3)^-1. Cosmological inflation can be realized, provided that N > or approx. 13. But the Brans-Dicke parameter is ω=(4epsilon)^-1, and ω > or approx. 500 means that N > or approx. 40. (author)