The radial mode of n extra compact dimensions (the radion b) can cause infl
ation in theories where the fundamental gravity scale M is smaller than the
Planck scale M-P. For radion potentials V(b) with a simple polynomial form
, to get the observed density perturbations, the energy scale of V(b) must
greatly exceed M similar to 1 TeV: V(b)(1/4)=M(V)similar to 10(-4) M-P. Thi
s gives a large radion mass and reheat temperature similar to 10(9) GeV, th
us avoiding the moduli problem. Such a value of M-V can be consistent with
the classical treatment if the new dimensions started sufficiently small. A
new possibility is that b approaches its stable value from above during in
flation. The same conclusions about M-V may hold even if inflation is drive
n by bulk matter fields rather than by the radion.