Several Conclusions on another site setting problem

Abstract

Let $S=\{A_1,A_2,\cdots ,A_n\}$ be a finite point set in m-dimensional Euclidean space $E^m$, and$\Vert A_i{A}_j\Vert$ be the distance between $A_i$ and $A_j$. Define $\sigma (S)=\sum _{1\le i<j\le n}\Vert A_i{A}_j\Vert$, $D(S)=\underset{1\le i<j\le n}{\max }\left\{\Vert A_i{A}_j\Vert \right\}$, $\omega (m,n)=\frac{\sigma (S)}{D(S)}$, $\sup \omega (m,n)=\max \left\{\frac{\sigma (S)}{D(S)}|S\subset E^m,\left|S\right|=n\right\}$. This paper proves that, for any point P in an n-dimensional simplex $A_1{A}_2\cdots A_{n+1}$ in Euclidean space, $\sum _{i=1}^{n+1}\Vert P{A}_i\Vert$ <= $\underset{i_t,j_t\in \{1,2,\cdots ,n+1\}}{\sup }\left\{\sum _{t=1}^n\Vert A_{i_t}{A}_{j_t}\Vert \right\}$ By using this inequality and several results in differential geometry this paper also proves that $\sup \omega (2,4)=4+2\sqrt{2-\sqrt{3}}$, $\sup \omega (n,n+2)$ >= $C_{n+1}^2+1+n\sqrt{2\left(1-\sqrt{\frac{n+1}{2n}}\right)}$.

10 pages