# Sum of Norms to Root of Function

Let $\alpha$ be a line defined by $\alpha(t): [0,1] \rightarrow \mathbb{R}^3$ and $x_1, \ldots, x_n \in \mathbb{R}^3$ be $n$ fixed points.

Let $A(t) = \sum_{i=1}^n \frac{1}{\|\alpha(t) - x_i\|_2}$ the sum of the distances from any point in $\alpha$ to all fixed $x_i$'s.

I want to take the problem of finding $A(t) \geq C$, for a fixed $C$, to finding $g(x) = 0$ for some $g$.

If it was $A(t) = C$ it seems it would be simple, but how to map that inequality to an equality involving the root of a function?