I consider points in the two-dimensional plane.

An equilateral triangle is a set of three points in the plane which are equidistant.

Suppose now I have $n$ points $x_1,...,x_n$. What is the configuration which minimizes: $$H(x):=\sum_{i,j} (a-\|x_i-x_j\|^2)^2$$ where $a$ is positive real number.

Clearly, if $n=3$, one recovers the equilateral triangle. Could you draw the solution for larger $n$ ? What is the value of $H_{min}$ as a function of $n$ ?

Thanks !