  We consider the spectral problem for a family of $N$ point interactions of
the same strength confined to a manifold with a rotational symmetry, a circle
or a sphere, and ask for configurations that optimize the ground state energy
of the corresponding singular Schr\"odinger operator. In case of the circle the
principal eigenvalue is sharply maximized if the point interactions are
distributed at equal distances. The analogous question for the sphere is much
harder and reduces to a modification of Thomson problem; we have been able to
indicate the unique maximizer configurations for $N=2,\,3,\,4,\,6,\,12$. We
also discuss the optimization for one-dimensional point interactions on an
interval with periodic boundary conditions. We show that the equidistant
distributions give rise to maximum ground state eigenvalue if the interactions
are attractive, in the repulsive case we get the same result for weak and
strong coupling and we conjecture that it is valid generally.
