On the optimal interpoint distance sum inequality

Let ({{left{x_{1}, dots, x_{n}right}subset mathbb{R}^2}}) be a set of points in the unit circle. It is shown that

$sumlimits^{n}_{i=1}{min_{j neq i}{left|x_{i} - x_{j}right|^2}}leq9,$

which is best possible and improves earlier results by Arpacioglu and Haas and Xia and Liu.