On the translative packing densities of tetrahedra and cuboctahedra

In 1900, as a part of his 18th problem, Hilbert asked the question to determine the density of the densest tetrahedron packings. However, up to now no mathematician knows the density δt(T) of the densest translative tetrahedron packings and the density δc(T) of the densest congruent tetrahedron packings. This paper presents a local method to estimate the density of the densest translative packings of a general convex solid. As an application, we obtain the upper bound in0.3673469⋯≤δt(T)≤0.3840610⋯, where the lower bound was established by Groemer in 1962, which corrected a mistake of Minkowski. For the density δt(C) of the densest translative cuboctahedron packings, we obtain the upper bound in0.9183673⋯≤δt(C)≤0.9601527⋯. In both cases we conjecture the lower bounds to be the correct answer.