2013 IEEE 54th Annual Symposium on Foundations of Computer Science (2008)

Oct. 25, 2008 to Oct. 28, 2008

ISSN: 0272-5428

ISBN: 978-0-7695-3436-7

pp: 189-198

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/FOCS.2008.50

ABSTRACT

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 $\Omega(\sqrt{d})$. Our main result is a construction with surface area $O(\sqrt{d})$, matching the lower bound up to a constant factor of $2\sqrt{2\pi/e} \approx 3$. The best previous tile known was only slightly better than the cube, having surface area on the order of $d$.

INDEX TERMS

Foams, Tiling, Parallel Repetition, Rounding

CITATION

Anup Rao,
Ryan OʼDonnell,
Avi Wigderson,
Guy Kindler,
"Spherical Cubes and Rounding in High Dimensions",

*2013 IEEE 54th Annual Symposium on Foundations of Computer Science*, vol. 00, no. , pp. 189-198, 2008, doi:10.1109/FOCS.2008.50SEARCH