In George W. Tokarsky's [*Polygonal Rooms Not Illuminable from Every Point*](http://doi.org/10.2307/2975263) (1995) it is stated that the problem



> 
> Is a polygonal region illuminable from at least one point in the region?
> 
> 
> 


was still open at the time. What is its current state? Has it been settled or is it still open?



Current state of Straus's illumination problem
As far as I know, the specific question you ask (entirely illuminable from at least one point) remains open.
(Tokarsky showed that placing a light in some spots can leave some points dark.)
But you may be interested to know that an old conjecture has been settled,
as I reported in
[this earlier question](https://mathoverflow.net/a/42007/6094):



> 
> Lelièvre, Monteil and Weiss have shown that if $P$ is a *rational* polygon,
>  for every $x$ there are at most finitely many $y$ not illuminated by $x$.
> 
> 
> 


Lelievre, Samuel, Thierry Monteil, and Barak Weiss. "Everything is illuminated." [arXiv:1407.2975](https://arxiv.org/abs/1407.2975) (2014).


