There is a conjecture by Pólya & Szegő (~1950, stated in p. 159 of their book [Isoperimetric Inequalties in Mathematical Physics](http://rads.stackoverflow.com/amzn/click/0691079889)) which is as follows:


"Of all $n$-gons of a fixed area, the regular $n$-gon minimizes the first Dirichlet eigenvalue."


Surprisingly, this is still open (to my knowledge) for the general case. The only settled cases are the triangles and the quadrilaterals (see [Henrot's survey](https://hal.inria.fr/hal-00115548/document)). 
 Is there any progress on the general case? 



A long-lasting conjecture of Pólya & Szegő
I'm pretty sure that this is still open for $n$-gons (with $n\geq 5$). As far as I know, basically no progress has been made since the original proofs for triangles/quadrilaterals. 


There have been some numerics as well as some refined inequalities for triangles. [This article](https://www.math.tecnico.ulisboa.pt/~pant/pp333_342.pdf) might point you towards some of these results.


Interestingly, a related conjecture of Pólya & Szegő is resolved "the regular $n$-gon has least logarithmic capacity among $n$-gons of a fixed area" <http://www.ams.org/mathscinet-getitem?mr=2052355>. 


