A small cosmological constant from a large number of extra dimensions

In this article, we consider the $4+n$ dimensional spacetimes among which one is the four dimensional physical Universe and the other is an n-dimensional sphere with constant radius in the framework of Lanczos-Lovelock gravity. We find that the curvature of extra dimensional sphere contributes a huge but negative energy density provided that its radius is sufficiently small, such as the scale of Planck length. Therefore, the huge positive vacuum energy, i.e. the large positive cosmological constant is exactly cancelled out by the curvature of extra sphere. In the mean time the higher order of Lanczos-Lovelock term contributes an observations-allowed small cosmological constant if the number of extra dimensions is sufficiently large, such as $n\approx 69$.