Summary: THE m≠CORE PROPERLY CONTAINS THE m≠DIVISIBLE
POINTS IN SPACE
School of Computer Science
Montreal, Canada, H3A 2A7
A point x in R d is an m - divisible point of a finite set of n points S Ő R d if x is contained in conv
S i , i = 1, . . . , m , for some partition S = S 1 » S 2 » . . . » 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 m≠core. 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≠