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It is shown that Einstein gravity in four dimensions with small cosmologicalconstant and small extra dimensions can be obtained by spontaneouscompactification of Lovelock gravity in vacuum. Assuming that the extradimensions are compact spaces of constant curvature, General Relativity isrecovered within certain class of Lovelock theories possessing necessarilycubic or higher order terms in curvature. This bounds the higher dimension tobe at least seven. Remarkably, the effective gauge coupling and Newton constantin four dimensions are not proportional to the gravitational constant in higherdimensions, but shifted with respect to their standard values. This effectopens up new scenarios where a maximally symmetric solution in higherdimensions could decay into the compactified spacetime either by tunneling orthrough a gravitational analogue of ghost condensation. Indeed, this is whatoccurs requiring both the extra dimensions and the four-dimensionalcosmological constant to be small.

General Relativity with small cosmological constant from spontaneous compactification of Lovelock theory in vacuum