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$\left\| {{A_i}{A_j}} \right\|$ be the distance between $A_i$ and $A_j$. Define $\sigma (S) = \sum\limits_{1 \le i < j \le n} {\left\| {{A_i}{A_j}} \right\|} $, $D(S) = \mathop {\max }\limits_{1 \le i < j \le n} \left\{ {\left\| {{A_i}{A_j}} \right\|} \right\}$, $\omega (m,n) = \frac{{\sigma (S)}}{{D(S)}}$, $\sup \omega (m,n) = \max \left\{ {\left. {\frac{{\sigma (S)}}{{D(S)}}} \right|S \subset {E^m},\left| S \right| = n} \right\}$. 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\limits_{i = 1}^{n + 1} {\left\| {P{A_i}} \right\|} $ <= $\mathop {\sup }\limits_{{i_t},{j_t} \in \{ 1,2, \cdots ,n + 1\} } \left\{ {\sum\limits_{t = 1}^n {\left\| {{A_{{i_t}}}{A_{{j_t}}}} \right\|} } \right\}$ 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 {{\textstyle{{n + 1} \over {2n}}}} } \right)} $.