To expand upon the title, and put it in clear terms, I phrase the problem thusly:
Consider the interior of any continuous, closed, non-self-intersecting curve in the plane. (I'm not sure if I'm stating the condition in its most rigorous format; I mean for the boundary curve to be a flat "loop" without gaps in it, and which does not intersect itself. The interior is a continuous, flat surface without holes in it. Examples: a star, a square, a circle, or an ellipse. Non-example: a donut, because of its hole.)
Now suppose you drop into the region in question an infinitesimal flashlight (mathematically represented by a point) which can shine in all directions. The flashlight's rays shine straight, and continue until they hit the boundary of the region. Thus the region will now be partially "shaded" by the light rays. In the case of a circle, no matter where you drop the flashlight, its rays will "shade" the whole circle. In the case of a star, you must drop the flashlight in the center to achieve this; if you drop it in one of the "wings" of the star, the rays will not fill up the whole star. In the case of a very curvy banana region, there is no point such that the light rays will fill up the space.
The previous examples show that we sometimes can fill up the whole space with light regardless of where we drop the flashlight, sometimes never, and sometimes it depends where you drop the flashlight (like in a star domain).
My question is this: Is there a mathematical way to describe, for any region like the ones I describe above, the point $\vec{p}$ (or set of points!) such that if I drop the flashlight at $\vec{p}$, I will maximize the area that will be shaded by light rays? A special case would be how to describe the point (or points) in a star domain (see above link) such that we could fill up the whole domain.
[Side note: I thought of this question today in multivariate calculus while we were discussing how to create a condition such that curl-free vector fields with singularities can be conservative fields (the answer was to restrict their domain to some star domain).]