Spherical cubes and rounding in high dimensions

What is the least surface area of a shape that tiles ℝ^d under translations by ℤ^d? Any such shape must have volume 1 and hence surface area at least that of the volume-1 ball, namely Ω(√d). Our main result is a construction with surface area Q(√d), matching the lower bound up to a constant factor of 2√2π/e ≈ 3. 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 ℝ^d by translations of any full rank discrete lattice Λ with surface area 2π ||V^-1||_fb, where V is the matrix of basis vectors of Λ, and ||·||_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 [11] in the discrete setting. Our construction gives an almost optimal noise-resistant rounding scheme to round points in ℝ^d to rectangular lattice points.