  It is found the exact solution of the Poisson equation for the
multidimensional space with topology $M_{3+d}=\mathbb{R}^3\times T^d$. This
solution describes smooth transition from the newtonian behavior $1/r_3$ for
distances bigger than periods of tori (the extra dimension sizes) to
multidimensional behavior $1/r^{1+d}_{3+d}$ in opposite limit. In the case of
one extra dimension $d=1$, the gravitational potential is expressed via compact
and elegant formula. It is shown that the corrections to the gravitational
constant in the Cavendish-type experiment can be within the measurement
accuracy of Newton's gravitational constant $G_N$. It is proposed models where
the test masses are smeared over some (or all) extra dimensions. In
10-dimensional spacetime with 3 smeared extra dimensions, it is shown that the
size of 3 rest extra dimensions can be enlarged up to submillimeter for the
case of 1TeV fundamental Planck scale $M_{Pl(10)}$. In the models where all
extra dimensions are smeared, the gravitational potential exactly coincides
with the newtonian one. Nevertheless, the hierarchy problem can be solved in
these models.
