{"id":"2a02d17c-8db9-4af4-8859-aa17165d5d29","arxiv_id":"2412.06344","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every sufficiently large dimension d, there exist smooth convex domains whose first nontrivial Neumann eigenfunction attains its maximum strictly inside the domain.","lead":"This paper constructs smooth convex domains in high dimensions where the first nontrivial Neumann eigenfunction has its largest value in the interior, not on the boundary. This disproves the long-standing hot spots conjecture for convex sets when the dimension is large enough.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The lifting from log-concave counterexamples to convex domains hinges on Theorem 2.6(C.4), and the Harnack/equicontinuity step for the singular slice potential is under-supported.","rationale":"The paper presents a coherent and ambitious construction, and the overall strategy is credible: build a log-concave counterexample, lift it to barrel domains, and remove the interior maximum by a shielding argument. I found no internal contradiction that directly disproves the main theorem. However, the single most load-bearing step is the uniform C0 convergence of the radial eigenfunctions, because the maximum ratio is not continuous under the weak L2 convergence that is more easily obtained. The reader's weakest-assumption analysis identified exactly this point: the Harnack machinery for the singular slice potential is asserted via a transfer argument and an admitted gap in the literature. My reading confirms that this is where the proof is least secure. Several other issues I noticed, such as sign inconsistencies in Proposition 2.19/2.20, the reversal of symmetry statements in the proof of Proposition 2.10, and the repeated V >= 0 versus V <= 0 mismatches, appear to be correctable typos rather than fatal flaws. The verification I propose directly targets the disputed transfer: if the slice semigroup can be shown to satisfy the needed Harnack and contraction estimates, then the rest of the lifting argument has enough support to carry the conclusion. Until that step is made fully explicit, the appropriate verdict remains conditional.","tokens_in":33870,"tokens_out":21805,"duration_ms":237252,"concrete_test":"For a one-dimensional base Omega = [-pi/2, pi/2] and V = 0, write the radial change explicitly: the slice semigroup is a one-dimensional diffusion on s in [0, sqrt(d)/2] with generator d_s^2 - (d/(sqrt(d)/2 - s)) d_s. Check analytically, or numerically for d = 2, 3, 4, whether the Li-Yau inequality and the contraction estimate eta_{P_t f} <= eta_f hold for this exact generator with constants independent of d. Also verify the semigroup identity (3.39) by applying both sides to a compactly supported smooth f and checking the implied Harnack inequality up to the singular endpoint s = sqrt(d)/2. If the identity or the uniform bounds fail there, Theorem 2.6(C.4) is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires passing from the heat-extension inequality (2.12) to the eigenfunction ratio (2.14), and this passage needs the uniform C0 convergence asserted in Theorem 2.6(C.4). That convergence is obtained through uniform boundedness and equicontinuity of the slice eigenfunctions phi_d^S on S_d(Omega,V). The only route supplied is the uniform Harnack inequality for the slice pair (S_d, sqrt(d) W_s^d), whose potential W_s^d is highly singular and not Lipschitz. Lemma 3.12 transfers the Li-Yau inequality from the barrel domain F_d via the radial identity (3.39), but that identity is asserted rather than proved, and Remark 3.13 explicitly admits that no self-contained proof was found. Moreover, Proposition 3.5 applies Proposition 3.8, the coupling contraction estimate eta_{P_t f} <= eta_f, directly to the slice semigroup; Proposition 3.8 is stated only for Lipschitz convex potentials with bounded gradient, which is false for sqrt(d) W_s^d. If either the identity (3.39) or the equicontinuity estimate fails near the singular endpoint, then (C.4) collapses, and the ratio of extrema of phi_d need not converge to the corresponding heat-extension ratio. The existence of interior hot spots for the constructed convex domains would then be unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":34097,"tokens_out":32623,"duration_ms":304722,"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":[{"comment":"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.","section":"§3.2.1, Lemma 3.12 and Eq. (3.39)"},{"comment":"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.","section":"§3.4, Lemma 3.16"},{"comment":"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.","section":"§3.3, Lemma 3.19"},{"comment":"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.","section":"§4, first paragraph"},{"comment":"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.","section":"§2.2, Lemma 2.11"}],"minor_comments":[{"comment":"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.","section":"§4, Proposition 2.10"},{"comment":"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.","section":"§2.4, Proposition 2.8"},{"comment":"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.","section":"§2.1, Theorem 2.6"},{"comment":"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.","section":"§3.2, Lemma 3.6"},{"comment":"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.","section":"§3.4, Lemma 3.16"}],"recommendation":"major_revision","confidential_remarks":"The paper attacks a well-known open problem and the high-level idea—lifting a log-concave counterexample to convex barrel domains—is appealing and worth pursuing. My main concern is that several central technical steps are either unproved or incorrect as stated: the Harnack transfer for the singular slice potential, the barrier function in Lemma 3.16, the radial SDE in Lemma 3.19, the similarity transformation in Section 4, and the heat-flow inequality in Lemma 2.11. These are load-bearing for Theorem 2.6 and hence for the main theorem. The author explicitly acknowledges one of these gaps in Remark 3.13. Because the issues are numerous and affect the proof of the central convergence result, I cannot recommend acceptance in the current form; a major revision would need to supply complete proofs or replace the faulty arguments."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat'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.\n\nSoft 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.\n\nFor 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.\n\nRecommendation: engage with it. Send to a careful referee, and do not let the typos mask the real gap.","headline":"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.","tokens_in":34630,"tokens_out":2602,"would_cite":true,"duration_ms":26161,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P15","35K05","35B50"],"pacs":[],"model":"deepseek-v4-flash","headline":"The hot spots conjecture fails for convex domains in high dimensions","keywords":["hot spots conjecture","Neumann eigenfunction","convex domain","high-dimensional Laplacian","log-concave measure","barrel domain","heat extension","Li-Yau Harnack inequality"],"falsifier":"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.","tokens_in":1865,"feed_emoji":"🔥","tokens_out":2072,"duration_ms":98740,"temperature":0.7,"pith_summary":"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.","feed_headline":"Hot spots conjecture fails for convex domains in high dimensions","feed_subtitle":"For all large enough d, the first Neumann eigenfunction's maximum is reached only inside the domain.","key_machinery":"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 $\\epsilon\\to 0$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"posed the convex-domain hot-spots conjecture that Theorem 1.2 disproves.","marker":"[Kaw85]"},{"why":"formulated the natural variants of the conjecture and gave early positive results for long thin domains, variants that the counterexample contradicts.","marker":"[BB99]"},{"why":"posed the two-hyperplane conjecture, another high-dimensional formulation that the counterexample refutes.","marker":"[Jer19]"},{"why":"provides the dimension-free Li-Yau Harnack inequality used to control eigenfunctions on barrel domains.","marker":"[LY86]"},{"why":"supplies the dimension-free Harnack inequality in the form applied to the slice domain.","marker":"[Wan06]"},{"why":"gave an earlier counterexample for planar domains with holes; the present work transfers the failure to convex domains in high dimension.","marker":"[BW99]"},{"why":"introduced the hot-spots constant $S_d$ and showed it is bounded; the construction implies $S_d > 1$ for large $d$.","marker":"[Ste23]"}],"fun_headline_variants":["Hot spots can hide inside high-dimensional convex bodies","Counterexample: Interior maxima in high-dim convex sets","High-dim convex sets break the hot spots conjecture","Hot spots move inward as dimension grows"],"cache_read_input_tokens":36736,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Hot spots can hide inside high-dimensional convex bodies","Counterexample: Interior maxima in high-dim convex sets","High-dim convex sets break the hot spots conjecture","Hot spots move inward as dimension grows"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000736,"raw_usage":{"total_tokens":3218,"prompt_tokens":801,"completion_tokens":2417,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":417,"completion_tokens_details":{"reasoning_tokens":2359}},"tokens_in":417,"tokens_out":2417,"duration_ms":17905,"temperature":1.0,"reasoning_tokens":2359,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T19:45:04.822169+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}