  It is shown that Einstein gravity in four dimensions with small cosmological
constant and small extra dimensions can be obtained by spontaneous
compactification of Lovelock gravity in vacuum. Assuming that the extra
dimensions are compact spaces of constant curvature, General Relativity is
recovered within certain class of Lovelock theories possessing necessarily
cubic or higher order terms in curvature. This bounds the higher dimension to
be at least seven. Remarkably, the effective gauge coupling and Newton constant
in four dimensions are not proportional to the gravitational constant in higher
dimensions, but shifted with respect to their standard values. This effect
opens up new scenarios where a maximally symmetric solution in higher
dimensions could decay into the compactified spacetime either by tunneling or
through a gravitational analogue of ghost condensation. Indeed, this is what
occurs requiring both the extra dimensions and the four-dimensional
cosmological constant to be small.
