I have an exercise to a give very rigorous prove to two observations of computation geometry. Obviously there are related. I've tried to prove them and wrote few ideas. Please take a look at them, and fell free to share with us your opinion.
Basic theory for problem 1:
A $q$ is a visible point in polygon $P$, if for every point $p \in P$, the segment $pq$ is entirely in $P$.
Problem 1:
Prove that $q$ is a visible point in polygon $P$, if every vertex $v_{i}$ of polygon $P$ can see $q$.
Proof of problem 1:
In convex polygon every point is defined as visible point, by definition of convex polygon, every segment between points of convex polygon is contained inside convex polygon.
In arbitrary polygon not every point is visible point. If we can separate nonconvex polygon to minimal number of convex polygons all visible point are contained on the line common to all convex polygons (if it exists). So obviously if all visible points placed on the common line they are visible from all vertex of initial nonconvex polygon.
The problem of point visibility related to the art gallery problem,on this case only one guard will be sufficient for the art gallery problem.
Basic theory for problem 2:
Kernel of polygon $P$ is a set of all visible point of $P$.
Problem 2:
Kernel of the polygon $P$ is the intersection of N half-planes.
Proof of problem 2:
To be more precise kernel is a intersection of left half-planes, with reference to a counterclockwise traversal of the boundary. Obviously, each half-plane is convex and the intersection of convex sets is again convex. Therefore by intersection of half-planes of the polygon we get a convex polygon which is contained in initial arbitrary polygon. By previous problem 1 we know that all points of convex polygon are visible points, therefore this convex polygon is a kernel.
I know the proofs are not enough strict and rigorous, fell free to modify them or write new one.
Thank you very much.