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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
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.2008.50
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