Point Location Problem in Polygon in Repetitive Mode for a Simple Polygon
First question:

By (my interpretation of the) definition, a "central point" exists if and only if the polygon is star-shaped; this condition can be tested in $Θ(n)$ time.  So the only thing you can do is run this algorithm, apply division wedge if you find a non-empty kernel, and apply another algorithm if you find an empty kernel.