Solution of a Plane Six Setting Sites Problem

We consider the optimal setting of n stations and solve the problem of n = 6 in the plane. Let S ⸦ E² be a set consisting of n points A₁, A₂, ..., A_n, d(A_i, A_j) stands for the distance between A_i and A_j, and $\sigma (S)=\sum _{1\le i\lt j\le n}d({A}_{i},{A}_{j})$, $d(S)=\mathop{min}\limits_{1\le i\ne j\le n}\{d({A}_{i},{A}_{j})\}\,,\,\mu (2,n)=\frac{\sigma (S)}{d(S)}\,(S\subset P,|S|=n)$, inf $\mu (2,n)=min\{\frac{\sigma (S)}{d(S)}|S\subset P,|S| =n\}$. In the paper, the results for $\text{inf}\text{\,\,{0.17em}}\mu (2,6)=13+4\sqrt{3}$ are obtained by several analytical methods, such as classifying, regional control and evaluating the boundary extremum.