Is there a name for this type of polygon?

Is there a name for a polygon in which you could place a light bulb that would light up all of its area? (for which there exists a point so that for all points inside it the line connecting those two points does not cross one of its edges)

Examples of "lightable" polygons:

Examples of "unlightable" polygons:

Solution 1:

Yes, those are called star-shaped polygons. They have numerous applications in mathematics, for example in complex analysis.

Solution 2:

More generally, such a set is a star domain, and is a trivial example of contractible space.

You may see this as a generalization of a convex set: indeed,

• $C\neq\emptyset$ is a convex domain if for every $x,y\in C$ you have that the line segment $\overline{xy}\subseteq C$ is contained in $C$; while
• $S\neq\emptyset$ is a star domain if there exists a point $y$ such that for every $x\in S$ it holds that $\overline{xy}\subseteq S$.

That is, in a star domain the point $y$ (there might be many such) is fixed. You can easily prove that a set $E\neq\emptyset$ is convex (actually simply connected) if and only if it is a star domain with respect to each center $y\in E$.

Solution 3:

I think you are looking for star domains. See also this related question on Mathoverflow.