We know that for any k:
Thus the minimum value for
would be $1$, but we must show that such a configuration can indeed exist on the plane. For this we will just pick three points $A_0,A_1,A_2$ such that they form an equilateral triangle and we get that our expression evaluates to $1$, so we are finished.
In the case where $m=0$ and $2$ points coincied our expression is undefined so it is not at a minimum there either.