Laugesen–Siudeja Conjecture 6.34 on Neumann eigenvalue means

Let $T$ be a nondegenerate planar triangle, and let $0<\lambda_1^N(T)\leq\lambda_2^N(T)$ be its first two positive Neumann Laplace eigenvalues. Define the harmonic and geometric means by $H(T)=\frac{2\lambda_1^N(T)\lambda_2^N(T)}{\lambda_1^N(T)+\lambda_2^N(T)}$ and $G(T)=\sqrt{\lambda_1^N(T)\lambda_2^N(T)}$ . Then: (i) among all triangles with a fixed perimeter, $H(T)$ is uniquely maximized by the equilateral triangle; and (ii) among all triangles with a fixed area, $G(T)$ is uniquely maximized by the equilateral triangle. Equivalently, if $E_P$ is the equilateral triangle with perimeter equal to that of $T$ , and $E_A$ is the equilateral triangle with area equal to that of $T$ , then $H(T)\leq H(E_P)$ and $G(T)\leq G(E_A)$ , with equality in either inequality only when $T$ is equilateral.

References

Primary source

arXiv

Additional references

Sharp Neumann eigenvalue means on triangles:an exact certificate proof and stability — arXiv — Guowei Dai, Yingxin Sun

Posted 6 days ago · edited 6 days ago

Progress summary

Refreshed 6 days ago

Claimed solved

A recent preprint claims to settle the conjecture, but its proof has not been independently checked.

Laugesen and Siudeja posed the relevant triangle inequalities in their 2009 work: the equilateral triangle should uniquely maximize two Neumann eigenvalue means under different normalizations. The tracked claim additionally gives a quantitative stability estimate.

Known results

Laugesen and Siudeja (2009) proved weaker area-normalized and mixed-normalization inequalities, with equality only for the equilateral triangle.
A 2022 result established a related first-eigenvalue maximization only for convex planar domains with two axes of symmetry; it does not settle the conjecture for all triangles.

Recent claimed proof

Guowei Dai and Yingxin Sun's preprint Sharp Neumann eigenvalue means on triangles: an exact certificate proof and stability claims the equilateral triangle uniquely maximizes both means and proves quantitative stability. This is an unrefereed preprint, and no independent mathematical assessment or verification was retrieved.

Current status (as of October 2026): The classical weaker inequalities and restricted-domain results are established, but the full conjecture is not independently verified; Dai and Sun's claimed resolution remains unverified.

Sources

No solutions have been posted yet.