Finite point-sets on S 2 with minimum distance as large as possible
This paper is the translation of the author's Habilitationsschrift [``Endliche Punktmengen auf der 2-Sphäre mit möglichst großem Minimalabstand'', Univ. Göttingen, Göttingen (1963); per bibl.] not published in a journal until now. The list of references has been supplemented. Let M be a set of \(k\geq 2\) points on the unit sphere. Let f(M) denote the minimum (spherical) distance between any two points of M, and let \(d_ k=\max f(M),\) where the maximum extends over all sets M of k points. The graph of M is the one-dimensional complex G on \(S^ 2\) whose vertices are the points of M and whose edges are the arcs PQ of length f(M) with \(P,Q\in M\). Two vertices that are joined by an edge of G are called neighbours. A set M of k points is said to be maximal if \(f(M)=d_ k\). A maximal set for a given number of vertices, which has the least possible number of edges, is called an optimal set. Similarly, if M and N are sets of k points, then N is said to be better than M if either \(f(N)>f(M)\), or \(f(N)=f(M)\) and the graph of N has fewer edges than the graph of M. Let Q be a vertex of G, and let P and R which are not neighbors of each other be neighbours of Q. It is assumed that the angular region determined by QP and QR does not contain a neighbour of Q. Let \(Q^*\) be the image of Q by reflection in the great circle PR. M is said to be irreducible if it cannot be improved by a displacement or a single reflection \(Q\to Q^*\) of a single point Q of M. Every optimal set is irreducible. Finally, let \(\phi_ 3(d)\) denote the interior angle of the equilateral triangle with side length d. The main results of this very remarkable paper are the following: (i) For each \(k=7\) to 12 there is, up to isometry, exactly one optimal k- set \(M_ k\). \(M_{12}\) is the set of vertices of a regular icosahedron; \(M_{11}\) is obtained from \(M_{12}\) by removing one vertex. (ii) If M is irreducible and \(f(M)=d\) such that \(90\circ >\phi_ 3(d)\geq 72\circ,\) then \(k\leq 12\) and the graph of M is isomorphic to one of 21 specified graphs. Six of these sets are \(M_ 7,\cdot \cdot \cdot,M_{12}\). Ten of the remaining sets depend on one or two parameters, the exact ranges of which are explicitly given. The sets are uniquely determined (up to isometry) for given values of their parameters. (iii) If \(7\leq k\leq 12,\) every maximal set is optimal. The problem of finding the maximal set for \(k=3\), 4, 6 and 12 was solved by \textit{L. Fejes Tóth} [Jahresber. Dtsch. Math.-Ver. 53, 66-68 (1943; Zbl 0028.07605)], for \(k=5\), 7, 8 and 9 by \textit{K. Schütte} and \textit{B. L. van der Waerden} [Math. Ann. 123, 96-124 (1951; Zbl 0042.166)] and for \(k=24\) by \textit{R. M. Robinson} [ibid. 144, 17-48 (1961; Zbl 0107.398)]. The present paper contains the solutions for the unsolved cases \(k=10\) and 11, confirming two conjectures established in the paper by Schütte and van der Waerden cited above [op. cit.]. The methods developed by the author are based on the clever combination of graph-theoretic and elementary geometric conclusions. The way of investigating graphs with one or more degrees of freedom deserves particular interest. Nevertheless, in the author's opinion, ``substantially different methods must be found for \(k>12''\). This seems to be true even for the unsolved problem concerning \(d_{13}.\) [Reviewer's remark: \textit{K. Böröczky} [Stud. Sci. Math. Hung. 18, No.2-4, 165-171 (1983; Zbl 0573.52020)] gave an alternative proof for the maximal set of 11 points.]
scientific article; zbMATH DE number 4050429
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