A finite subset of a Euclidean space is called an s-distance set if there exist exactly s values of the Euclidean distances between two distinct points in the set. In this paper, we prove that the maximum cardinality among all 5-distance sets in \(\mathbb {R}^3\) is 20, and every 5-distance set in \(\mathbb {R}^3\) with 20 points is similar to the vertex set of a regular dodecahedron.

中文翻译：

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距离集的十二面体猜想的证明

如果欧几里德空间的有限子集在集合中两个不同点之间存在精确的欧几里德距离值，则称为s距离集。在本文中，我们证明所有5-距离集之中的最大基数\（\ mathbb {R} ^ 3 \）是20，并且在每5-距离集\（\ mathbb {R} ^ 3 \）与20点类似于常规十二面体的顶点集。