Spherical Cubes and Rounding in High Dimensions

What is the least surface area of a shape that tiles Ropf
^d
under translations by Zopf
^d
? Any such shape must have volume 1 and hence surface area at least that of the volume-1 ball, namely Omega(radicd). Our main result is a construction with surface area O(radicd), matching the lower bound up to a constant factor of 2radic2pi/eap3. The best previous tile known was only slightly better than the cube, having surface area on the order of d. We generalize this to give a construction that tiles Ropf
^d
by translations of any full rank discrete lattice Lambda with surface area 2piparV
^-1
par
_fb
, where V is the matrix of basis vectors of Lambda, and par.par
_fb
denotes the Frobenius norm. We show that our bounds are optimal within constant factors for rectangular lattices. Our proof is via a random tessellation process, following recent ideas of Raz in the discrete setting. Our construction gives an almost optimal noise-resistant rounding scheme to round points in Ropf
^d
to rectangular lattice points.