I am wondering about the following problem: Given a polygon and the set of points $S$ inside it, what are the point(s) in $S$ from which the most area in $S$ is visible? Furthermore, what is the maximum visible area?
Here, I define $q$ to be visible from $p$ if the line segment between $p$ and $q$ is contained in $S$. This intends to capture the intuitive idea of what points in a room are visible when standing somewhere in the room. For example, in the figure below, the dark blue area is visible from point P, at the center of the top left quarter. The light blue area is not.
While the answer for any star-shaped domain is clear, finding the answer for arbitrary polygons seems difficult.
Question: How can we find the solution to the problem for a given polygon?