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There is a conjecture by Pólya & Szegő (~1950, stated in p. 159 of their book Isoperimetric Inequalities 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?

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    $\begingroup$ Is the regular $n$-gon even proved to be a local minimum for $\lambda_1$? The survey you cite doesn't cite such a result in the "case of polygons" sections (3.2, page 5). $\endgroup$
    – Noam D. Elkies
    Commented Apr 10, 2016 at 15:19
  • $\begingroup$ @NoamD.Elkies As far as I know, the Polya's proof is the only one out there. I have not seen any proof for even "local"minimality. I thought I had a good idea how to approach this problem but sadly my method is only good for quadrilaterals. $\endgroup$
    – BigM
    Commented Apr 10, 2016 at 16:08
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    $\begingroup$ Is this conjecture also connected in any way to the Selberg eigenvalue conjecture, or they just happen to share the word "eigenvalue"? $\endgroup$
    – Suvrit
    Commented Apr 10, 2016 at 16:48
  • $\begingroup$ At Survit, I looked at the link(Peter Sarnak's) paper on Selberg's conjecture.That is not in my realm however to my understanding Selberg's inequality is analog of general isopremeteric inequality stating among regions of same area circle(balls for higher dimensions)maximizes the first eigenvalue. M.Ruzhansky(Imperial College London) has some nice results pertaining iso. inequalities for Lie groups and connected Riemannian manifolds. $\endgroup$
    – BigM
    Commented Apr 10, 2016 at 20:59
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    $\begingroup$ Some recent numerical computations suggest that regular polygons are indeed minimizers of the first eigenvalue under area constraint: lama.univ-savoie.fr/~bogosel/faber_krahn_polygons.html I do not know any proof of the fact that regular polygons are local minimizers. It is possible to prove that they are critical points, i.e. the derivative of the first eigenvalue of a regular polygon is zero with respect to every perturbation of the vertices. $\endgroup$
    – Beni Bogosel
    Commented Apr 30, 2016 at 21:43

2 Answers 2

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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.

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    $\begingroup$ I remember Solynin gave a talk at our school a few years ago.Back then I wasn't interested in the topic and didnt really pay much attention.Well my loss. At any rate, I glanced at your link to their Annals paper and have to say that their result is very neat. $\endgroup$
    – BigM
    Commented Apr 11, 2016 at 4:00
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Together with D. Bucur we propose a strategy which could prove the conjecture for a fixed $n \geq 5$ using a finite number of certified numerical computations.

Our paper can be found here: On the polygonal Faber-Krahn inequality The key points are:

  • The second shape derivative is computed for a simple eigenvalue of a polygon. Classical shape derivative formulas do not apply directly due to the lack of regularity.

  • The Hessian matrix of $(x_i,y_i) \mapsto \lambda_1(P)$ is computed explicitly for general $n$-gons

  • For regular $n$-gons the eigenvalues of the Hessian matrix for $|P|\lambda_1(P)$ (scale-invariant version) are computed explicitly. They depend on solutions to 3 PDEs.

  • Positivity of eigenvalues of the regular $n$-gon is not obvious. We prove that four eigenvalues are $0$ (corresponding to translations, rotations, homotheties which do not change the scale invariant objective). If the remaining $2n-4$ eigenvalues are strictly positive, the local minimality holds.

  • We develop explicit numerical estimates for piecewise linear finite elements allowing to guarantee positivity of the remaining $2n-4$ eigenvalues on meshes which are fine enough.

  • In the final section we show that the optimal polygon should verify various bounds on geometric elements which can narrow the search space for admissible optimal polygons.

  • We also show that the eigenvalues of the Hessian are stable around the regular $n$-gon (fixing two vertices to eliminate the zero eigenvalues). Therefore there exists a quantifiable region of local minimality around the regular $n$-gon (not fully computed).

  • Thus the proof could be reduced to a finite number of verified computations: one computation for local minimality and for finding a local minimality neighborhood. A finite number of computations could exhaust the region between the regular $n$-gon and arbitrary $n$-gons verifying the geometric constraints (diameter, inradius, lower bound on smallest side, etc).

We also have a preprint concerning a complete proof for the local minimality of the regular $n$-gon for $n \leq 6$ using validated computing (interval arithmetics). The arxiv version can be found here

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