Let be $E$ the set of $n \choose 2$ straight line segments ( edges ).
The region $V(i) = \{ x \in E^2 \vert d(x, v_i) \leq d(x, v_j), j = 1,\dots,N\}$
is called Voronoi polygon associated with the vertex $v_i$.
How is the euclidean distance $d(x, v_j)$ between $x=(e_1, e_2)$ and a vertex $v_j$ defined?
The definitions come from
Lee, D. T.; Schachter, B. J., Two algorithms for constructing a Delaunay triangulation, Int. J. Comput. Inform. Sci. 9, 219-242 (1980). ZBL0441.68047.