Minimum of convex combination of squared distance functions

Question

Fixing $m$ points in $\mathbf{R}^n$, $c_1, \dots, c_m$, is it true that the minimum of $\sum_1^m \|x - c_k\|_2^2 w_k$ occurs at the minimum of $\sum_1^m \|x - c_k\|_2 w_k$, when $\sum_1^m w_k = 1$ ($w_k \in \mathbf{R}_+$)? Note that $\|\cdot\|_2$ means the usual Euclidean norm.

Answer 1

No. Take $w_k=1$ for all $k=1,2,\ldots, m$, and two points $c_1$ and $c_2$ extremely close together, and $c_3$ a unit distance away. Then $\sum_{i=k}^3 w_k \times d(x,c_k)$ $=\sum_{k=1}^3 d(x,w_k)$ is minimized at $x$ near $c_1,c_2$, but $\sum_{k=1}^3 w_kd^2(x,c_k)$ $=\sum_{k=1}^3 d^2(x,c_k)$ is minimized when $x$ is about a third of the way between $\{c_1,c_2\}$ and $c_3$.