On Mordellʹs Inverse Problem in Dimension Three Original Research Article

Given a latticeL=AZ3with determinantd(L)=det(A)>0, letκ(L)= sup{vol(P)/(8d(L)} where the supremum is taken over allo-symmetric parallelepipedsPwith faces parallel to the coordinate axes such thatP∩L={o}. We prove the following conjecture by Gruber: The absolute minimum of the functionκ(L) has value 8/7 cos2(π/7) cos(2π/7)=0.578416…=κ(L*) and is uniquely attained at the critical latticeL* of the star body x1x2x3less-than-or-equals, slant1. Moreover, this minimum is isolated: There exists a positiveηsuch thatκ(L)>κ(L*)+ηholds for any lattice which is not equivalent toL*. We also state a conjecture concerning higher minima ofκ(L).