The asymptotic convex hot spots constant: does $C_d\to\sqrt{e}$? (Conjecture 1.10 of de Dios Pont, Hsu and Taylor)

The hot spots ratio of a bounded domain $\Omega\subset\mathbb{R}^d$ — the ratio of the maximum of the first non-constant Neumann eigenfunction over $\Omega$ to its maximum over the boundary $\partial\Omega$ — quantifies the failure of Rauch's hot spots conjecture. de Dios Pont, Hsu and Taylor (arXiv:2508.16321, 2025) proved that the supremum $S_d$ of this ratio over all connected Lipschitz domains converges to $\sqrt{e}$ as $d\to\infty$, and posed (Conjecture 1.10) the same question with the supremum $C_d$ taken only over convex bounded domains — the class in which counterexamples to the hot spots conjecture were constructed in all sufficiently high dimensions by de Dios Pont (2024). The upper bound $\limsup_{d\to\infty}C_d\le\sqrt{e}$ is known, and the convex counterexamples give $\liminf_{d\to\infty}C_d>1$; the open core of the conjecture is whether convex domains are asymptotically as bad as general Lipschitz domains, i.e. whether $\liminf_{d\to\infty}C_d\ge\sqrt{e}$. No resolution or partial progress beyond the sandwich $1<\liminf C_d\le\limsup C_d\le\sqrt{e}$ is known; the conjecture is open.

Background

Let $\Omega\subset\mathbb{R}^d$ be a bounded connected domain and consider the Laplace operator with Neumann boundary conditions (the condition $\partial_\nu\psi=0$ on $\partial\Omega$, meaning an insulated boundary). Its eigenfunctions describe the modes of heat dissipation in the insulated body $\Omega$: the temperature fluctuation of a generic initial distribution around its mean is dominated, as $t\to\infty$, by the first non-constant Neumann eigenfunction $\psi_\Omega$ (an eigenfunction for the smallest positive Neumann eigenvalue). Rauch's hot spots conjecture (1974) asserted that $\psi_\Omega$ attains its maximum and minimum on the boundary $\partial\Omega$ — physically, that the hottest and coldest spots migrate to the insulated surface.

The conjecture is false in general: Burdzy and Werner constructed a bounded connected planar domain with holes whose first non-constant Neumann eigenfunction attains a strict maximum at an interior point, with later counterexamples having only a single hole. Positive results cover special classes — planar triangles, domains that are long and thin, planar convex sets with an axis of symmetry, and in higher dimensions cylinders and generalizations of the planar lip domains of Atar and Burdzy (notably the class treated by Kennedy and Rohleder). Because of these results, the conjecture was long believed for convex domains in all dimensions, until de Dios Pont (2024) constructed smooth centrally symmetric convex counterexamples in every sufficiently large dimension $d$. His method extends the problem from the uniform measure on convex sets to log-concave measures (probability densities of the form $e^{-V(x)}dx$ with $V$ convex, whose natural diffusion operator is $-\Delta+\nabla V\cdot\nabla$ with Neumann conditions).

A quantitative version of the question — how badly can the conjecture fail? — is measured by the hot spots ratio introduced by Steinerberger:

$\frac{\max_{x\in\Omega}\psi_\Omega(x)}{\max_{x\in\partial\Omega}\psi_\Omega(x)},$

which equals 1 exactly when the maximum is attained on the boundary. Steinerberger proved the ratio is uniformly bounded (by 60) over all bounded connected smooth domains, so the failure is never catastrophic; Mariano, Panzo and Wang then generalized the constant to bounded Lipschitz domains (domains whose boundary is locally the graph of a Lipschitz function) and proved dimension-dependent upper bounds with asymptotic upper bound $\sqrt{e}\approx1.6487$ as $d\to\infty$. Writing $S_d$ for the supremum of the ratio over all connected bounded Lipschitz domains in $\mathbb{R}^d$, de Dios Pont, Hsu and Taylor determined it exactly: $S_d$ equals the central value of an explicit radial profile $\eta_d$ on the unit ball (built from the first Dirichlet eigenfunction of the ball, hence from Bessel functions), no domain attains the supremum, and every extremizing sequence converges to a ball at a quantitative rate measured by the Fraenkel asymmetry (the normalized measure of the smallest symmetric difference with a translate of the volume-matched ball). In particular $\lim_{d\to\infty}S_d=\sqrt{e}$. The same paper shows the conjecture is asymptotically true in measure: the volume fraction of $\Omega$ on which $\psi_\Omega$ exceeds $\alpha$ times its maximal boundary value tends to 0, exponentially fast in $d$, for every fixed $\alpha>1$.

What remains unknown is the same extremal question restricted to convex domains. Let $C_d$ denote the supremum of the hot spots ratio over convex bounded domains $\Omega\subset\mathbb{R}^d$ only. The sharp Lipschitz result gives $\limsup_{d\to\infty}C_d\le\sqrt{e}$, since every convex domain is Lipschitz; de Dios Pont's construction gives $\liminf_{d\to\infty}C_d>1$ — it produces smooth centrally symmetric convex domains with a spectral gap (the first positive Neumann eigenvalue is simple) whose ratios converge to some constant strictly greater than 1, whose numerical value is not computed. Nothing between some constant above 1 and $\sqrt{e}$ is known. de Dios Pont, Hsu and Taylor conjecture that the convex supremum is asymptotically just as bad as the Lipschitz one: $\lim_{d\to\infty}C_d=\sqrt{e}$. Their plausibility argument is that the convex counterexample construction is radially symmetric in all but two dimensions, and that if the auxiliary function $\psi_{\Omega,V}$ entering the effective (log-concave) eigenvalue problem of that construction could be chosen with enough freedom, one would likely reach the $\sqrt{e}$ threshold; making this rigorous is exactly the open problem.

Problem Statement

Let $C_d$ denote the supremum, over all convex bounded domains $\Omega\subset\mathbb{R}^d$, of the hot spots ratio

$\frac{\max_{x\in\Omega}\psi_\Omega(x)}{\max_{x\in\partial\Omega}\psi_\Omega(x)},$

where $\psi_\Omega$ is a first non-constant eigenfunction of the Laplacian on $\Omega$ with Neumann boundary conditions (in case of eigenvalue multiplicity, any eigenfunction for the smallest positive Neumann eigenvalue). Determine whether

$\lim_{d\to\infty}C_d=\sqrt{e}.$

Since $\limsup_{d\to\infty}C_d\le\sqrt{e}$ is already known (every convex domain is Lipschitz, and the sharp Lipschitz-domain supremum satisfies $S_d\to\sqrt{e}$), the conjecture is equivalent to its lower-bound half: for every $\varepsilon>0$, does there exist $d_0$ such that for every $d\ge d_0$ there is a convex bounded domain $\Omega\subset\mathbb{R}^d$ with $\max_{x\in\Omega}\psi_\Omega(x)/\max_{x\in\partial\Omega}\psi_\Omega(x)\ge\sqrt{e}-\varepsilon$?

References

1. Jaume de Dios Pont, Alexander W. Hsu, Mitchell A. Taylor. Sharp bounds on the failure of the hot spots conjecture. arXiv:2508.16321, 2025. DOI: 10.48550/arXiv.2508.16321. https://arxiv.org/abs/2508.16321
2. Jaume de Dios Pont. Convex sets can have interior hot spots. arXiv:2412.06344, 2024. DOI: 10.48550/arXiv.2412.06344. https://arxiv.org/abs/2412.06344
3. Phanuel Mariano, Hugo Panzo, Jing Wang. Improved upper bounds for the Hot Spots constant of Lipschitz domains. Potential Analysis 59(2):771-787, 2023. arXiv:2110.03672. DOI: 10.1007/s11118-022-10001-4. https://arxiv.org/abs/2110.03672
4. Stefan Steinerberger. An upper bound on the hot spots constant. Revista Matemática Iberoamericana 39(4):1373-1386, 2023. arXiv:2106.03677 (2021). DOI: 10.4171/rmi/1350. https://arxiv.org/abs/2106.03677
5. Krzysztof Burdzy, Wendelin Werner. A counterexample to the hot spots conjecture. Annals of Mathematics 149(1):309-317, 1999. arXiv:math/9803030. DOI: 10.2307/121027. https://arxiv.org/abs/math/9803030
6. James B. Kennedy, Jonathan Rohleder. On the hot spots conjecture in higher dimensions. Transactions of the American Mathematical Society, Series B 13:108-131, 2026. arXiv:2410.00816 (2024). DOI: 10.1090/btran/245. https://arxiv.org/abs/2410.00816