Convex sets can have interior hot spots

Jaume de Dios Pont

REVIEW 5 major objections 5 minor 2 cited by

Convex sets can have interior hot spots

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Pith's one-line read The hot spots conjecture fails for convex domains in high dimensions

desk verdict A plausible and important counterexample to the convex hot spots conjecture in high dimensions, with one load-bearing Harnack step that is not yet fully proved. read the letter →

arxiv 2412.06344 v1 pith:ZSOQG4RT submitted 2024-12-09 math.AP math.SP

Jaume de Dios Pont This is my paper · ORCID

classification math.APmath.SP MSC 35P1535K0535B50

keywords hotspotsconjectureNeumanneigenfunctionconvexdomainhigh-dimensionalLaplacianlog-concavemeasurebarrelheatextensionLi-YauHarnackinequality

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The hot spots conjecture asserts that in any bounded convex domain, the first non-constant Neumann eigenfunction of the Laplacian should attain its maximum on the boundary. This paper constructs a sequence of smooth, centrally symmetric convex domains in dimensions tending to infinity for which the ratio of the maximum inside the domain to the maximum on the boundary converges to a number greater than one, so for large enough dimension the maximum is attained only in the interior. The construction first proves a log-concave analogue: a convex potential on a two-dimensional rectangle produces an eigenfunction whose associated heat extension develops an interior maximum. That counterexample is then lifted into high-dimensional barrel-shaped convex domains that simulate the measure $\exp(-V)$; eigenfunctions on these domains converge, as the dimension grows, to the heat extension, preserving the interior maximum. If the argument is correct, it disproves the convex-domain hot spots conjecture in all sufficiently high dimensions, along with its natural refinements.

What carries the argument

The load-bearing object is the barrel domain $F_d(\Omega,V) = \{(x,w)\in\Omega\times\mathbb{R}^{d+1} : |w| \le \tfrac12(\sqrt{d} - V(x)/\sqrt{d})\}$, a high-dimensional body whose radius profile encodes a convex potential $V$. The first eigenfunction on this barrel is radial in the $w$-variable because the Poincar\'e constant on high-dimensional balls is small; after a change of variables it becomes $\psi_d(x,t)$, and the main limit theorem says $\psi_d$ converges uniformly to the heat extension $h(x,t)$, the solution of $\partial_t h = \tfrac18(\Delta_x h + \lambda_{\Omega,V} h)$ with $h(x,0)=\phi_{\Omega,V}$ and Neumann conditions. The positive term $\lambda h$ breaks the parabolic maximum principle, so $h$ may develop an interior maximum. The paper engineers a rectangle $[-\pi/2,\pi/2]\times[-1,1]$ with a small convex potential $\epsilon V_q$ whose eigenfunction has a prescribed boundary profile, then attaches large wings that transport the profile so that the transported boundary values are lower than the values carried into the interior, forcing the maximum of $h$ to occur at interior points and to stay there as $\epsilon\to 0$.

What would settle it

Run a high-precision numerical computation of the first nontrivial Neumann eigenfunction on the explicit barrel domain $F_d(\Omega,V)$ for the rectangle and potential built in the construction, for increasing values of $d$, and test whether the interior-to-boundary maximum ratio exceeds $1$; if the slice eigenfunctions are not uniformly bounded or do not converge in $C^0$ to the heat extension $h_{\Omega,V}$, the ratio will not approach the predicted value and the lifting argument fails.

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Extended reading notes

Core claim

The central claim is that there exist smooth, centrally symmetric convex sets $\Omega_d \subset \mathbb{R}^d$ with a spectral gap whose first nontrivial Neumann eigenfunction $\phi_{\Omega_d}$ satisfies $\lim_{d\to\infty} \max_{\Omega_d}\phi_{\Omega_d}/\max_{\partial\Omega_d}\phi_{\Omega_d} = \lim_{d\to\infty} \min_{\Omega_d}\phi_{\Omega_d}/\min_{\partial\Omega_d}\phi_{\Omega_d} > 1$. Equivalently, for all sufficiently large $d$ the hot spot---the point where the eigenfunction is largest---lies in the interior, and the coldest point does as well. The proof obtains this by replacing the convex domain by a barrel over a fixed rectangle with a convex potential, showing that the barrel eigenfunctions converge to the solution of a forced heat equation, and then choosing the potential so that the heat equation's solution has a maximum away from the parabolic boundary. The maximum is preserved under the limiting procedure, giving a genuine convex-domain counterexample.

Load-bearing premise

The construction stands on the assumption that eigenfunctions on the barrel domains stay uniformly bounded and equicontinuous through a dimension-free Harnack-type bound for a non-Lipschitz, singular potential, a step the paper admits it could not find in the literature and replaces with an elementary argument; if that bound fails, the convergence of the barrel eigenfunctions to the heat extension breaks.

Editorial extensions

If this is right

The hot-spots conjecture for convex domains is false in all sufficiently large dimensions, including the natural variants stated for the problem in the cited literature.
Any theorem asserting that first nontrivial Neumann eigenfunctions on convex domains must peak on the boundary cannot hold without a dimension threshold.
The domains are symmetric under coordinate reflections, so the failure is not an artifact of asymmetry; it also rules out a natural two-axis-of-symmetry generalization in high dimensions.
The parabolic-limit mechanism, in which elliptic eigenfunctions on a high-dimensional body converge to a forced heat flow, gives a systematic route to constructing interior maxima.
No explicit dimension $d_0$ is computed; the proof is qualitative, and tracking constants directly is expected to give a threshold too large for practical computation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An unstated consequence is that the same barrel-lifting scheme could produce interior hot spots for other log-concave measures, for example Gaussian weights, by changing the base domain and potential.
Because the mechanism is a violation of the parabolic maximum principle, a similar construction might work for higher Neumann eigenvalues or for Robin eigenfunctions, where analogous reaction terms appear.
A numerical test would probably require an optimized version of the construction, since the proof's qualitative dimension threshold is expected to be enormous; seeing the phenomenon in moderate dimensions would be an independent check of the mechanism.
The wing-shielding idea suggests a general principle: large convex potential gradients can transport eigenfunction values outward while leaving a protected interior region where the maximum develops.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The headline: this paper likely kills the hot spots conjecture for convex domains in high dimension, and the construction is genuinely new. I think the claim is probably true, but the proof has a load-bearing step in Section 3 that is not fully supported, and I would want that checked before endorsing.

What's new and good: the barrel-domain construction F_d(Omega,V) that simulates a log-concave measure, the reduction to a forced heat equation, and the two-stage perturbation (small potential, then transport wings) to get initial data whose heat extension violates the parabolic maximum principle. That is a real idea, not a variant of the old counterexamples. The paper is also honest: Remark 3.13 admits the Harnack gap, and the author gives an explicit elementary path instead of hiding it. The overall structure is coherent and the result, if correct, is major.

Soft spots: the chain from the log-concave counterexample to convex domains goes through Theorem 2.6(C.4), the uniform C0 convergence of the slice eigenfunctions. That convergence rests on a uniform Harnack inequality for the slice potential sqrt(d) W_s^d, which is singular and not Lipschitz. Lemma 3.12 transfers the Li-Yau inequality from the barrel via the radial identity (3.39), but that identity is asserted, not proved. Separately, Proposition 3.8 is stated for Lipschitz convex potentials with bounded gradient, which the slice potential is not; applying it to the slice semigroup needs a justification that isn't there. The author says he tried to find a self-contained proof and couldn't; that's a red flag, though not a fatal one. These are technical gaps, not a conceptual flaw, but they are exactly where the proof would break if it breaks. There are also notation inconsistencies (the rectangle R is defined differently in Section 4 than in Section 2) and a few typos that should be cleaned.

For whom: this is for researchers in spectral geometry and PDE; it's a serious paper with a strong claim and a plausible path. It deserves a serious referee. I would send it out, with a specific instruction to focus on Section 3, Lemma 3.12, and the identity (3.39). If the Harnack step can be repaired, this is a publishable landmark. If not, the main theorem is unsupported.

Recommendation: engage with it. Send to a careful referee, and do not let the typos mask the real gap.

Referee Report

5 major / 5 minor

Summary. The paper claims to disprove the hot spots conjecture for convex domains in sufficiently high dimensions. The strategy is to replace a convex domain by a family of high-dimensional ``barrel'' domains F_d(Ω,V) that approximate the log-concave measure exp(-V)dx, to show that the first nontrivial Neumann eigenfunctions of these barrels converge to the solution h_{Ω,V} of a forced heat equation (Theorem 2.6), and then to construct a convex pair (Ω,V) for which this heat extension attains its maximum in the interior (Proposition 2.8). Combining these two ingredients yields domains Ω_d whose first eigenfunction has an interior maximum (Theorem 1.2). The proof is structured into a perturbation argument on a rectangle (Proposition 2.10), a transport argument in appended wings (Proposition 2.13), and a long technical proof of the eigenfunction convergence of barrel domains (Section 3).

Significance. If the proof were correct, the result would settle a well-known open problem, namely the hot spots conjecture for convex domains in high dimensions. The log-concave extension and the barrel construction are interesting and potentially influential ideas, and the paper contains a number of explicit, falsifiable claims rather than a purely abstract existence argument. The manuscript also gives credit to the technical difficulty of the Harnack step in Remark 3.13. However, the central convergence theorem is supported by several technical lemmas that are either asserted without proof or are incorrect as written; these gaps are load-bearing for the main result.

major comments (5)

[§3.2.1, Lemma 3.12 and Eq. (3.39)] The Li-Yau Harnack inequality for the slice pair (S_d, sqrt(d) W_s^d) is transferred from the barrel domain through the identity (P_t^{F_d} f_F)(x,w) = P_t^{S_d, sqrt(d) W_s^d} f(x, sqrt(d)/2 - |w|), but this identity is asserted rather than proved. Moreover, Theorem 3.7 and Proposition 3.8 are stated for Lipschitz convex potentials with bounded gradient, while W_s^d is not Lipschitz near the singular endpoint r = sqrt(d)/2. Lemma A2, the approximation argument cited for the non-Lipschitz case, explicitly assumes ||∇V||_{L^∞} < ∞. Since Proposition 3.5 and Theorem 2.6(C.4) rely on the uniform equicontinuity obtained from these estimates, the C^0 convergence of ψ_d to the heat extension is not established. Remark 3.13 acknowledges that no self-contained proof of the Harnack inequality in this setting was found; the manuscript does not supply the missing argument.
[§3.4, Lemma 3.16] The barrier function b in Eq. (3.48) does not satisfy the supersolution inequalities in (3.46). For the term 20t + 10d^{-1}|w|^2, one has ∂_t(20t) = 20 and Δ(10d^{-1}|w|^2) = 20(d+1)/d, so ∂_t b - Δb is negative, not nonnegative. More seriously, at the curved boundary |w| = (1/2)(√d - V(x)/√d), the bracket in Eq. (3.50) is approximately 10d^{-1} - exp(d/8 - |w|^2/(2+4t)), which at t=0 and |w| ≈ √d/2 is of order 10/d - 1 < 0; hence the required Neumann condition ∂_n b ≥ 0 fails. Thus the barrier estimate, and with it Lemma 3.14 and Corollary 3.15, are unsupported.
[§3.3, Lemma 3.19] The stochastic differential equation (3.67) is not the radial process of a (d+1)-dimensional Brownian motion. If H_s = 1 - 4|X_s|^2/d for a (d+1)-dimensional Brownian motion X_s, Itô's formula gives dH_s = -(4+4/d)ds - (8|X_s|/d)dB_s, not dH_s = -8(1+d^{-1})ds + d^{-1}(1-H_s)dB_s. The drift and diffusion coefficients in (3.67) are therefore incorrect. Since the bound E[exp(2λ_d s_*)] ≤ 1+t and the representation (3.69) are derived from this SDE, the temporal equicontinuity argument in Lemma 3.20, which is needed for Theorem 2.6(C.4), is not valid as written.
[§4, first paragraph] The similarity transformation used for the perturbation analysis is incorrect. For the operator Q_{ϵV} := -exp(ϵV/2)∇ exp(-ϵV)∇ exp(ϵV/2), a direct computation shows that exp(-ϵV/2) Q_{ϵV} exp(ϵV/2) is not equal to -Δ + ϵ∇V·∇; it contains additional first-order terms, including an extra drift term and a term proportional to ΔV. Therefore the claimed analyticity argument for differentiating the eigenpairs of -Δ + ϵ∇V·∇ is not justified. The first-order expansion itself is standard and could be proved by other means, but the proof as written does not establish it.
[§2.2, Lemma 2.11] The proof of Lemma 2.11 does not prove the decisive heat-flow inequality H_q(0,t) - H_q(1,t) > 0 for t ∈ [0,1]. It constructs a family q_δ and states that for δ small enough the hypotheses hold, but no argument is given for the heat-flow inequality for the limiting or perturbed data. This inequality is used in the second estimate of the proof of Theorem 1.2 (around Eq. (2.44)), so the existence of the required function q is not established.

minor comments (5)

[§4, Proposition 2.10] The domain of the rectangle is stated as R = [-π/2,π/2] × [-π/4,π/4] at the beginning of Section 4, whereas the rest of the paper uses R = [-π/2,π/2] × [-1,1]; the domain of the function q changes accordingly. Please reconcile the two conventions.
[§2.4, Proposition 2.8] Proposition 2.8 states that φ_{Ω,V} is antisymmetric in y and symmetric in x, but the constructed eigenfunction √(2/π) sin(x) + ϵβ(x,y) is antisymmetric in x and symmetric in y. One of the two statements appears to have the roles of x and y interchanged.
[§2.1, Theorem 2.6] Theorem 2.6 assumes a smooth convex pair with V ≥ 0, while Definition 2.3 and the construction require max V ≤ 0; the sign convention for V should be made consistent throughout.
[§3.2, Lemma 3.6] In the proof of Lemma 3.6, the displayed formula for W_s^d(x,r) uses the variable x in the logarithm where r is intended.
[§3.4, Lemma 3.16] In Eq. (3.50), the exponential factor is exp(d/8 - |w|^2/(2+4t)), which at t=0 and |w| ≈ √d/2 is of order one, not exp(-√d) as claimed in the subsequent estimate; this is related to the failure of the Neumann barrier condition noted above.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the counterexample is constructed explicitly and the heat-flow maximum is derived, not assumed.

full rationale

The derivation chain is not circular. The final counterexample is produced by explicit construction rather than by fitting a parameter to the target ratio: Lemma 2.11 constructs a function q with the required heat-flow property H_q(0,t) - H_q(1,t) > 0; Proposition 2.10 constructs a convex potential V_q and a perturbation beta by solving the linearized PDE, with beta(pi/2,y) - q(y) constant; Lemma 2.14 constructs the transport function g; and Propositions 2.17-2.20 show by maximum-principle and convergence arguments that the wing heat flow inherits the excess from this constructed initial data. The final inequalities (2.43) are consequences of the strong maximum principle applied to p0, whose initial data is the explicitly constructed Ex_g(beta), so the interior maximum is proven rather than assumed. The lifting step, Theorem 2.6, is also not circular: the eigenvalue convergence, radial symmetry, equicontinuity, and C0 convergence of the barrel eigenfunctions are established through spectral estimates, variational arguments, and the external Li-Yau Harnack inequality, not by presupposing the hot-spots conclusion. The limitation admitted in Remark 3.13, namely that the author could not find a self-contained proof of the Li-Yau inequality for the non-Lipschitz slice potential and therefore uses an elementary barrier path, is a mathematical-support gap in the equicontinuity argument, not a circular reduction of the conclusion to its premises. No load-bearing self-citation occurs, and the cited Harnack inequality is an independent external theorem with stated assumptions. Accordingly, there are no circular steps to report.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The construction introduces explicit mathematical objects (barrels, slice domains, wing potentials) but no unexplained postulates, free numerical parameters, or newly invented physical entities. The axioms listed are standard tools or structural assumptions of the convex-pair framework.

assumptions (5)

standard math Dimension-free Li-Yau Harnack inequality for convex domains and Lipschitz convex potentials.
Invoked in Theorem 3.7 and used throughout to establish uniform boundedness and equicontinuity of eigenfunctions on barrel and slice domains. Cited from Li-Yau [LY86] and Wang [Wan06].
domain assumption Convexity of domains and potentials throughout the construction.
The whole counterexample lives in the class of convex pairs (Ω,V); convexity is used for the reflected Brownian motion coupling, the variance of the heat kernel, and the maximum principle arguments.
domain assumption Spectral gap of the base rectangle (R, ϵV_q) for sufficiently small ϵ.
Proposition 2.10 asserts this perturbatively; the argument requires that the first eigenvalue of the unperturbed rectangle is simple and that the perturbation is small, so the gap persists.
standard math Smooth approximation of convex pairs (Lemma A2).
Used to extend results from smooth potentials to the explicit potentials in the construction, which may have limited regularity. The lemma provides convergence of eigenvalues and eigenfunctions.
standard math Poincaré constant bound for high-dimensional balls, λ_{B_1(R^{d+1})} ≥ d.
Lemma A3 is used in Lemma 3.2 to force radial symmetry of the barrel eigenfunctions, a crucial step in reducing to the heat-extension problem.

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Pith. "Pith review of Convex sets can have interior hot spots." pith.science (2026). https://pith.science/paper/ZSOQG4RT

@misc{pith2026241206344,
  author       = {Pith},
  title        = {Pith review of: Convex sets can have interior hot spots},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZSOQG4RT}},
  note         = {Machine review of arXiv:2412.06344}
}

abstract

The hot spots conjecture asserts that for any convex bounded domain $\Omega$ in $\mathbb R^d$, the first non-trivial Neumann eigenfunction of the Laplace operator in $\Omega$ attains its maximum at the boundary. We construct counterexamples to the conjecture for all sufficiently large values of $d$. The construction is based on an extension of the conjecture from convex sets to log-concave measures.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Sharp bounds on the failure of the hot spots conjecture
math.SP 2025-08 conditional novelty 8.0 of 10

The exact hot spots ratio is eta_d(0), extremizers do not exist, and it tends to sqrt(e) as d approaches infinity.
Hot spots in domains of constant curvature
math.SP 2025-08 unverdicted novelty 6.0 of 10

The hot spots conjecture holds for all non-acute geodesic triangles of constant negative curvature, with additional critical point and monotonicity results for other constant curvature triangles and polygons.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

On neumann eigenfunctions in lip domains

Rami Atar and Krzysztof Burdzy. On neumann eigenfunctions in lip domains. Journal of the American Mathematical Society , 17(2):243--265, 2004

work page 2004

[2] [2]

On the hot spots conjecture of J

Rodrigo Ba \ n uelos and Krzysztof Burdzy. On the hot spots conjecture of J . R auch. Journal of functional analysis , 164(1):1--33, 1999

work page 1999

[3] [3]

The hot spots problem in planar domains with one hole

Krzysztof Burdzy. The hot spots problem in planar domains with one hole. Duke Mathematical Journal , 129:481--502, 2005

2005

[4] [4]

A counterexample to the `` H ot spots" conjecture

Krzysztof Burdzy and Wendelin Werner. A counterexample to the `` H ot spots" conjecture. Annals of mathematics , pages 309--317, 1999

work page 1999

[5] [5]

Monotone properties of the eigenfunction of neumann problems

Hongbin Chen, Yi Li, and Lihe Wang. Monotone properties of the eigenfunction of neumann problems. Journal de Math \'e matiques Pures et Appliqu \'e es , 130:112--129, 2019

work page 2019

[6] [6]

Parabolic theory as a high-dimensional limit of elliptic theory

Blair Davey. Parabolic theory as a high-dimensional limit of elliptic theory. Archive for Rational Mechanics and Analysis , 228(1):159--196, 2018

work page 2018

[7] [7]

The two hyperplane conjecture

David Jerison. The two hyperplane conjecture. Acta Mathematica Sinica, English Series , 35(6):728--748, 2019

work page 2019

[8] [8]

Euclidean triangles have no hot spots

Chris Judge and Sugata Mondal. Euclidean triangles have no hot spots. Annals of Mathematics , 191(1):167--211, 2020

2020