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MR0109315 (22 #201) 52.00 Birch, B. J. On 3N points in a plane. Proceedings of the Cambridge Philosophical Society 55 (1959), 289–293. The following theorem is proved in this note. Theorem 1: Given 3N points in a plane, we can divide them into N triads such that, when we form a triangle with the points of each triad the N triangles will all have a common point. The proof is given on the basis of three lemmas and two corollaries. The first two lemmas are the fixed-point theorem for n-space and Caratheodory’s (n + 1)point theorem. Lemma 3 is as follows, where E is the unit n-ball: Let a mass distribution in E be defined by an integrable density function ρ(x); then we can find a point r inside E so that every closed half-space with r on its boundary will contain at least 1/(n+ 1) of the total mass. The first corollary states that Lemma 3 holds if the mass-distribution is not continuous, and the second corollary is as follows: Let Y be a finite set consisting of M points in n-space, and suppose that M > (n + 1)R. Then there is a point common to all the closed half-spaces which contain at least (M −R) points of Y . {The author states that he believes Lemma 3 is new. Actually the most important new feature concerns the number and dispositions of closed half-spaces containing an “optimal” portion of the mass stated in the proof. For the Lemma 3 itself, it may be interpreted as a corollary of the finite point-mass problem which is a straight-forward generalization to n-space of a theorem stated and proved in Jaglom and Boltjanski [Konvexe Figuren, VEB Deutscher Verlag, Berlin, 1955; MR0079789; p. 16], where they also observe that the continuous case is special. To prove Theorem 1, then, the author could have applied his proof directly to the theorem stated by Jaglom and Boltjanski. However, the procedure used and the application of the method to other problems of optimization are of further interest.} P. C. Hammer From MathSciNet, August 2022