  We study the problem for the optimal charge distribution on the sites of a
fixed Bravais lattice. In particular, we prove Born's conjecture about the
optimality of the rock-salt alternate distribution of charges on a cubic
lattice (and more generally on a d-dimensional orthorhombic lattice).
Furthermore, we study this problem on the two-dimensional triangular lattice
and we prove the optimality of a two-component honeycomb distribution of
charges. The results holds for a class of completely monotone interaction
potentials which includes Coulomb type interactions. In a more general setting,
we derive a connection between the optimal charge problem and a minimization
problem for the translated lattice theta function.
