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It is shown that if N points are placed on the unit sphere in Euclidean 3-space so that the sum of the distances which they determine is maximal, then the distance between any two points is at least 2/3N. Results for sums of Ath powers of distances are also given. The following problem is open: How should one place N points p1, * PN on the surface of the unit sphere U in E3 (i.e. on x + y + z ) so that the sum of the distances which they determine is maximal? Many closely related problems have been studied; see for example [1]-[10]. Here we show that the points must be "well spaced" in the sense that no two can be closer than a distance of 2/3N. In fact, we will show a bit more. Let lpi Pjl denote the Euclidean distance between pi and pi. Theorem. Let 0 [4/A(2 +X)] 1/(2-A)N1/(2-X). We can improve the constant here, but it is more important to improve the exponent of N. Since min 1Pi pjl >N-"2 for 0<A< 2. It is interesting to observe that spherical distributions of points with maximal spherical distance sums are not necessarily well spaced [6], [8]. For the proof of the Theorem let PI, I * * X PN maximize the sum Received by the editors November 20, 1973 and, in revised form, February 12, 1974. AMS (MOS) subject classifications (1970). Primary 52A25, 52A40. Copyright ( 1975, American Mathematical Society

Spherical distributions of $N$ points with maximal distance sums are well spaced