TY - GEN

T1 - Spherical cubes and rounding in high dimensions

AU - Kindler, Guy

AU - Rao, Anup

AU - O'Donnell, Ryan

AU - Wigderson, Avi

PY - 2008

Y1 - 2008

N2 - 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 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 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.

AB - 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 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 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.

UR - http://www.scopus.com/inward/record.url?scp=57949091182&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2008.50

DO - 10.1109/FOCS.2008.50

M3 - Conference contribution

AN - SCOPUS:57949091182

SN - 9780769534367

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 189

EP - 198

BT - Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008

T2 - 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008

Y2 - 25 October 2008 through 28 October 2008

ER -