Need help in understanding the argument in a proof of polygon triangulation

Question

I cannot really understand the highlighted argument. I have feeling that the statement is correct, but I do not understand the argument that proves it.

Thank you!

Answer 1

A line segment is convex, and in particular this implies that it has some nice properties with respect to convex functions. Here the distance to line $(uw)$ is a convex function. If you consider segment $[vv']$, every point on that segment is further away from line $(uw)$ than point $v'$, but closer than point $v$. In fact this holds for every line segment not parallel to line $(uw)$ (by convexity).

Suppose for a contradiction that there exists an edge of $\mathcal P$ that intersects segment $[vv']$, let $[ab]$ be that edge and $i$ the point of intersection. Then $i$ is farther away than $v'$ from line $(uw)$, yet closer than $a$ or $b$. So either $a$ or $b$ is farther away from line $(uw)$ than $v'$. That vertex must also belong to the interior of triangle $uvw$ because segment $[ab]$ cannot cross $[uv]$ or $[vw]$. So the existence of that vertex contradicts the definition of $v'$.