# 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:

• I fail to see how the second is not "lightable"? The first one also seems to just be lightable but it depends on the exact size. Dec 6 '15 at 17:08
• @Runemoro: the second seems to consist of two overlapping triangles. As a triangle is convex, it is entirely lit by any point inside it, so putting the point in the tiny area where the triangles overlap should lit the entire figure. Dec 6 '15 at 19:07
• Similarly the first seems to be three triangles, if the top of the lower triangle is high enough to be inside the right triangle. Dec 6 '15 at 19:08
• For the "lightable" polygons, the possible locations you can put the lightbulb and still light the whole interior is called the kernel. Dec 6 '15 at 19:50
• They are the polygons for which the Art Gallery number is 1. Dec 6 '15 at 22:50

• $C\neq\emptyset$ 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 $y\in E$.