 
Summary: Spherical Cubes and Rounding in High Dimensions
Guy Kindler #
Weizmann Institute
guy.kindler@weizmann.ac.il
Ryan O'Donnell +
Carnegie Mellon University
odonnell@cs.cmu.edu
Anup Rao #
Institute for Advanced Study
arao@ias.edu
Avi Wigderson
Institute for Advanced Study
avi@math.ias.edu
June 16, 2009
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 volume1 ball,
namely# # d). Our main result is
a construction with surface area O( # d), matching the lower bound up to a constant factor of 2 p 2#/e # 3.
The best previous tile known was only slightly better than the cube, having surface area on the order of d.
