Finite point-sets on S 2 with minimum distance as large as possible
type: Item
instance of: scholarly article
title: Finite point-sets on S 2 with minimum distance as large as possible
zbMATH Open document ID: 0644.52007
DOI: 10.1016/0012-365X(86)90002-6
author: B. George
published in: Discrete Mathematics
publication date: 1986-01-01T00:00:00Z
review text: 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 t…
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 05B40
Mathematics Subject Classification ID: 52A37
Mathematics Subject Classification ID: 52C17
zbMATH DE Number: 4050429
zbMATH Keywords: minimum distance
zbMATH Keywords: optimal k-set
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4050429