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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:

enter image description here

Examples of "unlightable" polygons:

enter image description here

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    $\begingroup$ 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. $\endgroup$ Dec 6 '15 at 17:08
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    $\begingroup$ @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. $\endgroup$ Dec 6 '15 at 19:07
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    $\begingroup$ Similarly the first seems to be three triangles, if the top of the lower triangle is high enough to be inside the right triangle. $\endgroup$ Dec 6 '15 at 19:08
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    $\begingroup$ For the "lightable" polygons, the possible locations you can put the lightbulb and still light the whole interior is called the kernel. $\endgroup$
    – Silverfish
    Dec 6 '15 at 19:50
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    $\begingroup$ They are the polygons for which the Art Gallery number is 1. $\endgroup$
    – Pål GD
    Dec 6 '15 at 22:50
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Yes, those are called star-shaped polygons. They have numerous applications in mathematics, for example in complex analysis.

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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 $y\in E$.

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

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