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 dimensions 4 through 7 and 9 through 12. We also give a different three-point bound for lattice packing and conjecture that this second bound is sharp in dimension 4.
Joint work with Henry Cohn and David de Laat