 
Summary: Approximation and Exact Algorithms for MinimumWidth Annuli and Shells \Lambda
Pankaj K. Agarwal y Boris Aronov z Sariel HarPeled x Micha Sharir 
Abstract
Let S be a set of n points in R d . The ``roundness'' of S can
be measured by computing the width ! \Lambda (S) of the thinnest
spherical shell (or annulus in R 2 ) that contains S. This pa
per contains four main results related to computing ! \Lambda (S):
(i) For d = 2, we can compute in O(n log n) time an annulus
containing S whose width is at most 2! \Lambda (S). (ii) For d = 2
we can compute, for any given parameter '' ? 0, an an
nulus containing S whose width is at most (1 + '')! \Lambda (S),
in time O(n log n + n='' 2 ). (iii) For d – 3, given a pa
rameter '' ? 0, we can compute a shell containing S of
width at most (1 + '')! \Lambda (S) in time O
i n
'' d log
i diam(S)
! \Lambda (S)''
jj
or O
