3维欧氏空间中确定不同距离的最优点集  

Optimal Point Sets Determining Few Distinct Distances in Three-dimensional Euclidean Space E^3

作　　者：王俊 张玉琴 WANG Jun;ZHANG Yuqin(Center for Applied Mathematics,Tianjin University,Tianjin,300072,P.R.China;School of Mathematics,Tianjin University,Tianjin,300354,P.R.China)

机构地区：[1]天津大学应用数学中心,天津300072 [2]天津大学数学学院,天津300354

出　　处：《数学进展》2020年第3期375-384,共10页Advances in Mathematics（China）

基　　金：NSFC(No.11801410).

摘　　要：本文拓展Erdős和Fishburn的工作,研究在3维欧氏空间中确定不同距离的有限点集结构.令f(k)表示可以在3维欧氏空间中放置点的最大数目,使得这些点恰好确定k个不同的距离.我们证明了f(1)=4,f(2)=6和f(3)≥12,并给出了k=1,2,3时对应的最优点集构型.In this paper,we generalize the work of Erd\H{o}s and Fishburn to study the structure of finite point sets that determine few distinct distances in three-dimensional Euclidean space E3.Let f(k)denote the maximum number of points that can be placed in E3 to determine exactly k distinct distances.We show f(1)=4,f(2)=6 and f(3)≥12.Also,we characterize the optimal configuration for k=1,2,3.

关 键 词：Erdős不同距离猜想 最优构型 有限点集 3维欧氏空间 

分 类 号：O157.3[理学—数学]