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The problem of finding the densest packing of tetrahedra was first suggested by Hubert [3, p. 319]. Minkowski [4] attempted to find the densest lattice packing of tetrahedra, but his result is invalid due to the incorrect assumption that the difference body of a regular tetrahedron was a regular octahedron. A lower bound for the maximum density of such a packing has been given by Groemer [ l ] as 18/49. The purpose of this paper is to announce the proof that 18/49 is in fact the maximum possible density. We shall use the term convex body to mean any compact, convex set in three dimensions with nonempty interior, and lattice to mean the collection of all points (vectors) mA+nB+pC, where m, n, p range over all integers and A, B, C are three fixed linearly independent vectors. If J is a convex body and A is a lattice such that, when x and y are distinct points of A, the bodies x + J and y + J have no interior points in common, then the collection of bodies A + J = {x + J : x £ A } is said to form a lattice packing. If A(A) = |det(^4, Bf C)\, and Vol(J) represents the volume of J , then the density of the packing is defined to be Vol(J)/A(A). Minkowski [4] showed that A + J is a lattice packing if and only if there are no points of A other than the origin in the interior of the difference body J —J = {x— y:x, yÇzJ}When the latter condition holds, A is said to be admissible for the difference body. A lattice is critical for a difference body if it is admissible and if no other admissible lattice has a smaller determinant. I t follows that the problem of finding the densest lattice packing for a given convex body J is equivalent to that of finding a critical lattice for J—J. The following lemmas are also from Minkowski's paper.

The densest lattice packing of tetrahedra