|
A Radon partition of a subset
P of Rd is a pair {A,B} satisfying (i) A ∪ B = P, (ii) A ∩ B = ∅ and (iii) conv A∩
conv B≠∅. The sets A and B are called components of the partition. The
theorem of Radon says that for any P ⊂ Rd having at least d + 2 elements,
there exists a Radon partition. When P is in general position with exactly
d + 2 elements, the Radon partition is unique; furthermore, a pair of points
of P lie in the same component if and only if they are separated by the
hyperplane through the remaining d points. A generalization of this result
is
Theorem 1. Let P be a set of n ≧ d + 2 points of Rd in general position, and let
S ⊂ P have k elements. Then S is contained in a component of some Radon
partition of P if and only if (i) k ≦n−d − 1; or, (ii) if k ≧n−d, then conv S∩ aff
(P ∼ S)≠∅.
With the notion of a primitive partition, a useful “reduction” is obtained.
Theorem 2. Every Radon partition of P extends a primitive partition.
Finally, a new characterization of the unique Radon partition mentioned above is
given by
Theorem 3. Let P be a set of d + 2 points in general position in Rd which do not
lie on a common sphere. Then a pair of points in P lie in the same component
of the unique Radon partition if and only if both of them are inside (or
both outside) the respective (d − 1)-spheres determined by the other d + 1
points.
|
