 
Summary: THE mCORE PROPERLY CONTAINS THE mDIVISIBLE
POINTS IN SPACE
David Avis*
School of Computer Science
McGill University
3480 University
Montreal, Canada, H3A 2A7
July 1991
ABSTRACT
A point x in R d is an m  divisible point of a finite set of n points S Ì R d if x is contained in conv
S i , i = 1, . . . , m , for some partition S = S 1 È S 2 È . . . È S m of S. Let D m (S) denote the set of m
divisible points of S. We say x is in the m  core of S if each closed half space containing x also contains
at least m points of S. Let C m (S) denote the set of points in the mcore. Reay, Sierksma and others have
conjectured that convD m (S) = C m (S). In this note we give a counterexample for n = 9, d = 3, m = 3. The
conjecture in known to be true for d = 2.
* Research supported by the Natural Science and Engineering Research Council Grant number A3013 and the F.C.A.R.
Grant number EQ1678.
2
Let S be a finite set of n points in R d . A point x in R d is an m  divisible point of S if x is con
