Vol. 36, No. 1, 1971

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Characterizations of Radon partitions

William Ray Hare, Jr. and John Willis Kenelly

Vol. 36 (1971), No. 1, 159–164

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 nd 1; or, (ii) if k nd, then conv Saff (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.

Mathematical Subject Classification
Primary: 52.34
Received: 29 January 1970
Published: 1 January 1971
William Ray Hare, Jr.
John Willis Kenelly