Spherical Cubes and Rounding in High Dimensions

What is the least surface area of a shape that tiles Rd under translations by Zd ? 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 O( √ d), matching the lower bound up to a constant factor of 2 p 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 Rd 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 Rd to rectangular lattice points.

• Type: Conference
• Year: 2008
• Where: FOCS
• Authors: Guy Kindler, Ryan O'Donnell, Anup Rao, Avi Wigderson
• Keywords: Constant Factors, FOCS 2008, Rectangular Lattice, Surface Area, Theoretical Computer Science