  We define three-point bounds for sphere packing that refine the linear
programming bound, and we compute these bounds numerically using semidefinite
programming by choosing a truncation radius for the three-point function. As a
result, we obtain new upper bounds on the sphere packing density in dimension 4
through 7 and 9 through 16. We also give a different three-point bound for
lattice packing and conjecture that this second bound is sharp in dimension 4.
