Suppose I have a star-shaped region in the plane with a particular point marked. The marked vertex is in the kernel of the star.
In the image the left most point is marked in blue. Yes I drew this in paint-- don't judge me. Just by looking it would appear that the largest (area) convex region inside this star that has the marked point as a vertex is the following yellow subset:
Is there a mathematical way (algorithm) to compute this region? I don't see another way to do it rather than inspection. That is, by sight I can remove the points that stick way out into pointy regions, but I want a way to do this systematically (not by sight) using the coordinates of all the vertices.
This "largest convex region inside this star that has the marked point as a vertex" actually is not a well-defined concept. Imagine a non-convex quadrangle $ABCD$ with left-most vertex $A$ as marked vertex, and a reflex angle at $C$.
How would you define your largest convex region in that case?