Why do we live in a 4D world: Can cosmology, black holes and branes give   an answer?
  We derive the general form of the cosmological scalar field potential which is compatible both with the existence of black holes and p-branes related to string/M theory and with multidimensional inflationary cosmology. It is shown that the scalar potential alters non-trivially from dimension to dimension yet always obeys one single equation where the number of spacetime dimensions is a free parameter. Using this equation we formulate an eigenvalue problem for the dimensionality parameter. It turns out that in the low-energy regime of sub-Planckian values of the inflaton field, i.e., when the Universe has cooled and expanded sufficiently enough, the value four arises as the largest admissible (eigen)value of this parameter. 
Konstantin G. Zloshchastiev