Spherical cubes and rounding in high dimensions

Guy Kindler*, Anup Rao, Ryan O'Donnell, Avi Wigderson

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

10 Scopus citations

Abstract

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.

Original languageAmerican English
Title of host publicationProceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008
Pages189-198
Number of pages10
DOIs
StatePublished - 2008
Externally publishedYes
Event49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008 - Philadelphia, PA, United States
Duration: 25 Oct 200828 Oct 2008

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428

Conference

Conference49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008
Country/TerritoryUnited States
CityPhiladelphia, PA
Period25/10/0828/10/08

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