I have a set $\{C_1, C_2, \ldots, C_N\}$ of $N$ nonintersecting closed piecewise linear curves on the Euclidean plane. For every point $x \in \mathbb{R}^2$ we say it belongs to a territory serviced by $C_i$ if $C_i$ is closest curve to $x$.

The boundaries of territories serviced by the $C_i$'s are clearly polygonal lines themselves (???). I am looking for the algorithm to find the vertices of each boundary.

Thanks, Vladimir