  We investigate lattice energies for radially symmetric, spatially extended
particles interacting via a radial potential and arranged on the sites of a
two-dimensional Bravais lattice. We show the global minimality of the
triangular lattice among Bravais lattices of fixed density in two cases: In the
first case, the distribution of mass is sufficiently concentrated around the
lattice points, and the mass concentration depends on the density we have
fixed. In the second case, both interacting potential and density of the
distribution of mass are described by completely monotone functions in which
case the optimality holds at any fixed density.
