We evaluate the one-loop effective potential for the Einstein–Hilbert action coupled to a nonlinear sigma model in a Kaluza–Klein background space M 4 × S N (M 4 is the four-dimensional Minkowski space–time and S N is the N-dimensional sphere) for odd N. The computation is made in the harmonic and in the light cone gauges. The radius of compactification for some N was found.