Consider the space $\mathbb{R}^2.$ We are given $n$ mutually different points $x_1,..,x_n \in \mathbb R^2.$
We can then introduce expressions
$$f_i(x_1,...,x_n) = n^2 \vert x_i \vert^2 + \sum_{j \neq i } \frac{1}{\vert x_j-x_i \vert^2 }.$$
I would like to know if there exists an explicit constant $c>0$, independent of the points, (as large as possible) such that $$ n^2+ \sum_{i=1}^n f_i(x_1,...,x_n) \ge c n^3.$$
I will do two cases by hand to give you a feeling:
$n=1$: This case is clear, as
$$1+ f_1(x_1) \ge 1 =c1^3$$ with $c=1.$
$n=2$: This case is already more tricky, however
$$4 + f_1(x_1,x_2) + f_2(x_1,x_2) =4+ 4(\vert x_1 \vert^2+\vert x_2 \vert^2) + \frac{2}{\vert x_1-x_2 \vert^2}$$ and thus using the parallelogram identity we find $$ 4 + f_1(x_1,x_2) + f_2(x_1,x_2)=4+ 2 (\vert x_1-x_2 \vert^2 + 1/\vert x_1-x_2 \vert^2) + 2 \vert x_1+x_2 \vert^2.$$
Now, we may use that $t+1/t \ge 2$ for $t > 0$ to infer that
$$ 4 + f_1(x_1,x_2) + f_2(x_1,x_2)\ge 4+ 4 = 2^3.$$
So somehow one could conjecture that $c=1$ is possible, but I don't know whether this is true in general.