For a positive integer $d$, a set of points in $d$-dimensional Euclidean space is called almost-equidistant if for any three points from the set, some two are at unit distance. Let $f(d)$ denote the largest size of an almost-equidistant set in $d$-space. It is known that $f(2)=7$, $f(3)=10$, and that the extremal almost-equidistant sets are unique. We give independent, computer-assisted proofs of these statements. It is also known that $f(5) \ge 16$. We further show that $12\leq f(4)\leq 13$, $f(5)\leq 20$, $18\leq f(6)\leq 26$, $20\leq f(7)\leq 34$, and $f(9)\geq f(8)\geq 24$. Up to dimension $7$, our work is based on various computer searches, and in dimensions $6$ to $9$, we give constructions based on the known construction for $d=5$. For every dimension $d \ge 3$, we give an example of an almost-equidistant set of $2d+4$ points in the $d$-space and we prove the asymptotic upper bound $f(d) \le O(d^{3/2})$.
On a special class of equidistant sets in the Euclidean space
Institute of Mathematics, University of Debrecen（德布勒森大学数学研究所）；ELKH-DE Equations, Functions, Curves and their Applications Research Group（ELKH-DE 方程、函数、曲线及其应用研究组）；Department of Mathematics and Informatics, Ferenc Rakoczi II Transcarpathian Hungarian University（费伦茨·拉科齐二世外喀尔巴阡匈牙利大学数学与信息系）
Three-dimensional antipodal and norm-equilateral sets
我们刻画了至少允许六个或至少七个等距点的三维空间。特别地，我们证明了$\R^3$上允许六个等距点的$C^\infty$范数的存在性，这反驳了Lawlor和Morgan（1994，Pacific J. Math \textbf{166}，55- 83）的一个猜想，并给出了$\R^3$上某些一致凸范数的具有六个区域的能量最小化锥的存在性.另一方面，$\R^3$上的可微范数不允许有7个等距点。证明中的一个关键要素是对所有三维对跖集的分类。我们还将所得结果应用于几个三维凸体的接触数。
