正方形内四点和五点问题的精确值

The Precision Values about the Problems of Four and Five Points in a Square

在单位边长正方形内ABCD内任意放置n个点P1 ,P2 ,……Pn,记入 (P1 ,P2 ,……Pn) =min{|pipj|i≠j,i,j=1,2 ,… ,n},λ n =sup{λ( p1 ,p2 ,…pn) | p1 ,p2 ,…pn 是正方形ABCD内任意n点 }.文献 [1]中指出λ 3 ～λ 1 0 的精确值尚未确定 ,[2 ]中证明了λ 3 =6 - 2 ,本文进一步证明了λ 4=1和λ 5 =22 .

Let p 1,p 2,…p n be n different points in a regular 4-gon ABCD with unit Length,λ(p 1,p 2,…p n)=min{｜p ip j｜i≠j,i,j=1,2,…n},λ=sup{λ(p 1,p 2…p n)｜p 1,p 2,…,p n are any n different points in ABCD}.The precision Values of λ * 3～λ * 10 are unknown from [1],and λ * 4=6-2 was proved in [2],In this paper,I prove that λ * 4=1 and λ * 5=22.