I have the following problem, related to finding a point in space that minimizes the maximum weighted distance between the point itself and $n$ circles.
Given $a_1, \dots, a_n \in \mathbb{R}$, $ {\bf c}_1, \dots, {\bf c}_n \in \mathbb{R}^d$ and $r_1, \dots, r_n > 0$, where $d \gg n$,
$$ \min_{{\bf x} \in \mathbb{R}^d} \max_{1 \leq i \leq n} a_i \left( \| {\bf x} - {\bf c}_i \|_2 + r_i \right)^2 $$
Can this problem be solved analytically? Can we at least prove that the solution is a convex combination of ${\bf c}_1, \dots, {\bf c}_n$?
I can prove that the solution is a convex combination for $d=1$ and arbitrary $n$ by using the fact that the extreme points of the convex hull of ${\bf c}_1, \dots, {\bf c}_n$ is the $\max_{i} {\bf c}_i$ and $\min_{i} {\bf c}_i$. However, I don't think my argument can be extended to an arbitrary $d$.