TITLE: A long-lasting conjecture of Pólya & Szegő
QUESTION [31 upvotes]: There is a conjecture by Pólya & Szegő (~1950, stated in p. 159 of their book Isoperimetric Inequalties in Mathematical Physics) 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). 
 Is there any progress on the general case?

REPLY [12 votes]: 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 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.