{"id":"28b93ec4-9dbd-453f-9c0d-99da721bc993","arxiv_id":"2505.01066","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For -1<p<1 and q sufficiently close to n, the near-isotropic Lp dual Minkowski problem on the sphere has a unique solution, with a sharp C0 estimate; the even case covers -1<p<q<min{n,n+p}.","lead":"Mathematicians prove that, for a family of geometric surface-weighting formulas on spheres, the corresponding shape reconstruction problem has a unique solution whenever the target data is nearly constant. This resolves a long-studied uniqueness question for negative parameters in the Lp dual Minkowski problem and supplies the sharp a priori bounds that make the proof work.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"In the proof of Theorem 1.6(a), equation (72) is said to contradict Theorem 4.5(ii), but that theorem is stated for cone-volume measures; converting (72) to a cone-volume density inserts the unproved factor h^p_{K∞}, so the contradiction is not established for p≠0.","rationale":"The main theorem is plausible and the inverse-function-theorem part is sound: the linearized operator Δ_{S^{n-1}}+(n-p)I is invertible for p∈(-1,1), and the regularity bootstrap in Section 7, though sketched, follows standard Caffarelli–Schauder arguments. The weak point is the compactness step. The cited Theorem 4.5, as stated, concerns cone-volume measures, and the proof's bridge from the limiting Lp surface-area measure to a cone-volume measure is missing exactly the constancy of h^p. This is not a criticism of the authors' intent: if Saroglou's original theorem is for S_p with p∈(-1,1), the proof goes through, and if not, an elementary Minkowski-space support argument appears to repair the gap. Because the written proof contains this missing justification, preserving the existing CONDITIONAL verdict is appropriate rather than accepting the proof as fully verified or rejecting the theorem outright.","tokens_in":31428,"tokens_out":21394,"duration_ms":235170,"concrete_test":"Check the precise statement of the cited non-existence result in Saroglou [85] (arXiv:2109.06545) and Böröczky–Saroglou [14]. If it is stated for S_{p,K} with p∈(-1,1), then the application is valid and the concern is only a misstatement of Theorem 4.5; if it is only for V_K (p=0), recompute the cone-volume density in (72) as ψ=(β/n)h_{K∞}^p and show that a non-round limit K∞ gives nonconstant ψ, so Theorem 4.5(ii) does not apply. Then test the repair: derive a contradiction directly from (72) by noting h_{K∞}≥θ>0, hence S_{K∞} is supported in the proper subsphere L∩S^{n-1}, which is impossible for a bounded full-dimensional body by the Minkowski condition that surface-area measure is not contained in a closed hemisphere. If that repair goes through, the theorem is salvageable; if not, Theorem 1.6(a) lacks a proof for p≠0.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing step is the contradiction in the proof of Theorem 1.6(a) in Section 4, in the λm→1 case. After the anisotropic blow-up, equation (72) yields that S_{p,K∞} = β H^{k-1} restricted to L∩S^{n-1}, and the text states that 'this fact contradicts Theorem 4.5(ii)'. However, Theorem 4.5(ii) is a theorem about the cone-volume measure V_K: its density ψ on L∩S^{n-1} cannot be constant. To pass from S_{p,K∞} to V_{K∞}, one uses dV_K = (1/n) h_K^p dS_{p,K}, the q=n case of (50). The induced cone-volume density is ψ = (β/n) h_{K∞}^p. For p=0 this is constant, but for p∈(-1,1)\\setminus{0} constancy is equivalent to h_{K∞}|_{L∩S^{n-1}} being constant, which the proof never shows and which is not a consequence of the preceding limit formulas. Since the main novelty of Theorem 1.3 is exactly p∈(-1,0), the C0 estimate for p≠0 is not justified as written. The bounded-density variant for n=3,4 survives because only positivity of the density is needed there, but the λm→1 case is essential and requires constancy.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies uniqueness in the Lp qth dual Minkowski problem near the isotropic case. The main result, Theorem 1.3, states that for n ≥ 2, α ∈ (0,1), and p ∈ (-1,1), there exists ε0 > 0 such that if |q - n| < ε0 and f ∈ C^α(S^{n-1}) is ε0-close in C^α to the constant function 1, then the equation d~Cp,q,K = f dH^{n-1} has a unique solution K ∈ K_o^n in the Alexandrov sense, and h_K|S^{n-1} is a positive C^{2,α} solution of (4). The proof combines new C0 estimates (Theorems 1.6 and 1.7) with a local inverse function theorem and a convergence argument in Section 7. The paper also proves Theorem 1.5, a near-isotropic uniqueness result for even solutions when -1 < p < q < min{n, n+p} and q > 0.","tokens_in":31716,"tokens_out":13431,"duration_ms":151890,"significance":"If correct, Theorem 1.3 is a substantial advance: it settles near-isotropic uniqueness for the full range p ∈ (-1,1), including the previously open case p ∈ (-1,0), and it extends the result from Lp surface-area measure to Lp qth dual curvature measures with q close to n. The C0 estimate with the sharp threshold p > -1, complementing the counterexamples of Jian-Lu-Wang, is an important contribution in its own right. The paper is generally carefully written, with detailed derivations of the anisotropic blow-up estimates and explicit statements of the dependence of constants. The main reservation concerns a gap in the application of Theorem 4.5 in the proof of Theorem 1.6(a), as detailed below; the bounded-density variant for n = 3, 4 is not affected by that gap.","major_comments":[{"comment":"The statement 'This fact contradicts Theorem 4.5(ii)' is not justified for p ≠ 0. Equation (72) identifies the Lp surface-area measure S_{p,K∞} with βH^{k-1} on L ∩ S^{n-1}, whereas Theorem 4.5(ii) is a statement about the cone-volume measure V_{K∞}. The conversion via formula (50) with q = n gives dV_{K∞} = (1/n) h_{K∞}^p dS_{p,K∞}, so the induced cone-volume density on L ∩ S^{n-1} is (β/n) h_{K∞}^p. For p ∈ (-1,1) \\ {0}, constancy of this density is equivalent to constancy of h_{K∞} on L ∩ S^{n-1}, and that constancy is not proved anywhere in the argument. Since this contradiction is exactly what rules out lim_m diam K_m = ∞ in Theorem 1.6(a), the C0 estimate for p ≠ 0 lacks support as written. The gap is repairable: one could either prove that h_{K∞}|_{L ∩ S^{n-1}} is constant, or replace the appeal to Theorem 4.5(ii) by the standard fact that the surface-area measure of a full-dimensional convex body cannot be supported on a proper subsphere. For the bounded-density case n = 3,4, the same conversion is harmless because only positivity/boundedness below of the cone-volume density is needed.","section":"Section 4, equation (72)"},{"comment":"The passage from L∞ convergence of h_{K_m} to 1 to C^{2,α} convergence is asserted with a reference to 'Caffarelli's regularity theory and Schauder estimates (similar to the arguments in [23, Proof of Lemma 4.1])' and no details are given. This step is load-bearing because it is used to contradict (119) and to conclude that the solution is positive and C^{2,α}. The manuscript should either provide the bootstrap argument or state and prove the precise compactness/regularity lemma that applies under the hypotheses |q_m - n| → 0, f_m → 1 in C^α, together with the a priori bounds from Theorem 1.6. Section 7 is introduced as a sketch, but Theorems 1.3 and 1.5 are the main theorems of the paper, so this is a substantive omission rather than a purely expository one.","section":"Section 7, proof of Theorem 1.3"}],"minor_comments":[{"comment":"The word 'Holder' should be spelled 'Hölder'.","section":"Abstract"},{"comment":"'in the letter case' appears to be a typo for 'in the latter case'.","section":"Introduction, page 2"},{"comment":"The proof of Claim 4.1 refers to equation '(56)' when verifying estimate (54); the cross-reference should be to the displayed estimate being proved, or the numbering should be corrected.","section":"Section 4, Claim 4.1"},{"comment":"The proof of Theorem 1.5 is described as 'similar to that of Theorem 1.3' and then concludes via [21]; since the parameter regime and limiting equation differ, one or two sentences explaining how Theorem 1.7 and [21] combine would improve clarity.","section":"Section 7"}],"recommendation":"major_revision","confidential_remarks":"The contested step in Section 4 relies on Theorem 4.5, a non-existence result from the authors' earlier work (Saroglou [85] and Böröczky-Saroglou [14]). This is a legitimate tool, but the application is not straightforward because of the conversion between Lp surface-area measure and cone-volume measure. The paper's central novelty—the extension to p ∈ (-1,0)—is clear and not a repackaging of prior results; the main issue is the repair of the C0-estimate proof and the regularity bootstrap in Section 7."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper is worth serious attention, but the advertised p∈(-1,0) uniqueness is not fully supported by the written proof. The C^0 estimate in Theorem 1.6 has a load-bearing gap that the authors will need to close.\n\nWhat is genuinely new: Theorem 1.3 extends near-isotropic uniqueness from p∈[0,1) to the full range p∈(-1,1), and the p<0 part was explicitly open in earlier work. The C^0 estimate is new and the sharpness at p=-1 is a nice addition. Theorem 1.5 is a clean result for the even case. The overall architecture is sensible: Lemma 3.2 gives the regularity, the anisotropic blow-up yields the C^0 estimate, and uniqueness follows from an inverse-function theorem near the constant solution plus a compactness argument. The derivation of the limit measure (equations (71)-(72)) is careful, and the use of KLS ellipsoids is standard.\n\nNow the soft spot. In the proof of Theorem 1.6, case λ_m→1, equation (72) shows that S_{p,K∞} = β H^{k-1} restricted to L∩S^{n-1}. The text says this contradicts Theorem 4.5(ii). But Theorem 4.5(ii) is a statement about cone-volume measures, not L_p surface area measures. To convert, you use dV = (1/n) h^p dS_p (equation (50) with q=n). The induced cone-volume density is then (β/n) h_{K∞}^p. That is constant only if p=0 or if h_{K∞} happens to be constant on L∩S^{n-1}. The proof never shows the latter, and nothing in the preceding limit formulas forces it. For p=0 the step works, and for the n=3,4 bounded-λ variant the weaker contradiction only needs positivity, so that part likely survives. But for p≠0 in the λ_m→1 case, the contradiction as written is not established. Since the main selling point of Theorem 1.3 is exactly p∈(-1,0), this is not a minor typo.\n\nEverything else I looked at holds up. Section 7 is admittedly a sketch and delegates the final C^{2,α} bootstrap to standard arguments; I would want that spelled out in a published version, but it is not the main issue. The self-citation load is notable but not circular—Theorem 4.5 is a real prior result—it is just misapplied here.\n\nWho should read this: anyone working on L_p dual Minkowski problems or on uniqueness near isotropic data. It deserves a serious referee. My recommendation is major revision: either fix the contradiction in Theorem 1.6(a), or restrict the statement of Theorem 1.3 to p=0 if the gap cannot be closed. I would not cite the p∈(-1,0) claim in its current form.","headline":"The p∈(-1,0) uniqueness result is promising but the proof's C^0 estimate has a real gap: the key contradiction only works for p=0.","tokens_in":32319,"tokens_out":7461,"would_cite":false,"duration_ms":78818,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A20","35J96"],"pacs":[],"model":"deepseek-v4-flash","headline":"For p in (-1,1), near-isotropic solutions of the Lp dual Minkowski problem are unique.","keywords":["Lp dual Minkowski problem","near-isotropic uniqueness","C0 estimate","convex body","Lp surface area measure","dual curvature measure","weak measure solution"],"falsifier":"Fix $n=2$, $p\\in(-1,0)$, $q=2$, and take $f_\\varepsilon=1+\\varepsilon\\,Y_{1,0}$ on $S^1$, with $Y_{1,0}$ the first spherical harmonic and $\\varepsilon$ small. Theorem 1.3 predicts that $d\\tilde C_{p,2,K}=f_\\varepsilon\\,dH^1$ has exactly one convex solution, with support function $1-\\varepsilon(2-p)^{-1}Y_{1,0}+O(\\varepsilon^2)$; a second solution branch, a bifurcation at $\\varepsilon=0$, or a sequence of bodies with nearly uniform $\\tilde C_{p,2,K}$ whose diameters tend to infinity would refute the theorem.","tokens_in":31218,"feed_emoji":"🔵","tokens_out":16378,"duration_ms":143247,"temperature":0.7,"pith_summary":"The paper proves that the near-isotropic $L_p$ dual Minkowski problem has a unique solution: if the exponent $p$ lies between $-1$ and $1$, the dual exponent $q$ is sufficiently close to the space dimension $n$, and the prescribed density $f$ on the sphere is Hölder-close to the constant function $1$, then exactly one convex body (in the weak measure sense) has that density as its $L_p$ dual curvature measure. This closes the previously open negative range $-1<p<0$ for the classical $L_p$ Minkowski problem (the case $q=n$), where solutions had been known to be unique in the exactly isotropic case but non-unique in general. The result matters because it converts a sharp $C^0$ bound — bodies with nearly uniform $L_p$ dual curvature cannot be arbitrarily large or arbitrarily flat — into a global uniqueness statement, and it exposes $p=-1$ as an optimal barrier: for $p\\le -1$ the same $C^0$ estimate is known to fail.","feed_headline":"Near-isotropic Minkowski uniqueness proved for all p in (-1,1)","feed_subtitle":"A sharp C0 estimate settles uniqueness for near-spherical bodies, including the previously open negative range","key_machinery":"The load-bearing object is the $L_p$ $q$th dual curvature measure $\\tilde C_{p,q,K}$ on the sphere, defined by $d\\tilde C_{p,q,K}=h_K^{-p}\\,d\\tilde C_{q,K}$, where $\\tilde C_{q,K}(\\omega)=\\int_{\\alpha_K^*(\\omega)}\\rho_K^q\\,dH^{n-1}$ records how much of the boundary of $K$ is seen radially from directions in $\\omega$. The argument runs on two mechanisms. First, a $C^0$ estimate (Theorem 1.6) is proved by anisotropic blow-up: if a body with nearly uniform $\\tilde C_{p,q,K}$ became unbounded, an affine rescaling would force its $L_p$ surface-area measure to converge to a measure supported on a lower-dimensional subspace with constant density there, and a non-existence result (Theorem 4.5) rules that configuration out; the same estimate also rules out collapse to zero volume. Second, an inverse-function step at the constant solution uses the linearized operator $L\\varphi=\\Delta_{S^{n-1}}\\varphi+(n-p)\\varphi$, which is invertible on Hölder spaces because the spherical Laplacian has eigenvalues $k(k+n-2)$ and because $n-p>0$ for $p<1$. The $C^0$ estimate upgrades this local diffeomorphism into a global uniqueness theorem.","core_discovery":"The central claim is Theorem 1.3: for $n\\ge 2$, $\\alpha\\in(0,1)$ and $p\\in(-1,1)$, there is an $\\varepsilon_0>0$ such that whenever $|q-n|<\\varepsilon_0$ and $\\|f-1\\|_{C^\\alpha}<\\varepsilon_0$, the measure equation $d\\tilde C_{p,q,K}=f\\,dH^{n-1}$ has a unique weak-measure solution $K\\in K_o^n$, and $u=h_K|_{S^{n-1}}$ is a positive $C^{2,\\alpha}$ solution of the corresponding Monge–Ampère equation. The proof fuses three ingredients: an optimal $C^0$ estimate (Theorem 1.6) bounding the diameter and volume of any body whose $L_p$ dual curvature is close to the spherical measure; a compactness argument showing such bodies converge to the unit ball, using the known uniqueness of the exactly isotropic solution; and local invertibility of the map $(q,u)\\mapsto(q,(\\|\\nabla u\\|^2+u^2)^{(q-n)/2}u^{1-p}\\det(\\nabla^2 u+uI))$ at $(n,1)$, whose linearization is $\\Delta\\varphi+(n-p)\\varphi$ and is invertible precisely because $n-p>0$. In the even category the same scheme yields Theorem 1.5: uniqueness for $-1<p<q<\\min\\{n,n+p\\}$ with $q>0$.","pith_inferences":["The same local-to-global scheme suggests a general recipe: for any family of curvature measures whose linearization at the constant solution is invertible and for which a $C^0$ estimate rules out escaping to infinity or degenerating to a lower-dimensional body, near-isotropic uniqueness should follow by the same argument.","The proof's blow-up analysis ties near-isotropic uniqueness to a purely structural statement — no constant $L_p$ surface-area density can be supported on a proper subspace — so any counterexample for $p\\in(-1,1)$ would have to produce a degenerate limit whose density is not constant, bypassing that structural obstruction.","The optimality of the barrier $p=-1$ suggests a testable prediction: no norm on the perturbation $f-1$ with the same scaling behavior can restore near-isotropic uniqueness for $p\\le -1$; a different mechanism would be needed there."],"forward_implications":["When $q=n$ and $p\\in(-1,0)$, the theorem settles the near-isotropic $L_p$ Minkowski problem, the last open range after $p\\in[0,1)$ had been handled previously.","For every $p\\in(-1,1)$ and $q$ close enough to $n$, the problem is well-posed near $f\\equiv 1$: the solution is unique, its support function is positive and $C^{2,\\alpha}$, and it is $C^{2,\\alpha}$-close to the constant function $1$ when $f$ is $C^\\alpha$-close to $1$.","The range $-1<p<1$ is optimal for the $C^0$ estimate that carries the proof: for $p\\le -1$ with $q=n$, examples exist of bodies with arbitrarily small volume whose $L_p$ surface-area density lies between two fixed positive constants.","In the even (origin-symmetric) setting, uniqueness holds for the larger window $-1<p<q<\\min\\{n,n+p\\}$, $q>0$, so the even theory is not confined to the near-isotropic $q\\approx n$ regime.","Because the proof identifies any limiting body as the unit ball, the uniqueness statement is accompanied by a continuity statement: as $f$ tends to $1$ in $C^\\alpha$, the unique solution $K$ tends to the unit ball in the Hausdorff metric."],"supporting_citations":[{"why":"Supplies the uniqueness of the exactly isotropic solution, used to identify the limiting body in the compactness argument as the unit ball.","marker":"[15]"},{"why":"Establishes near-isotropic uniqueness for $p\\in[0,1)$ and, together with [85], provides the non-existence theorem that rules out degenerate blow-up limits.","marker":"[14]"},{"why":"Proves the non-existence result for constant $L_p$ surface-area density on a lower-dimensional subspace, the contradiction tool in the $C^0$ estimate.","marker":"[85]"},{"why":"Provides the proof pattern for Section 7: local invertibility of the linearized map plus a $C^0$ estimate yields global uniqueness near the constant solution.","marker":"[23]"},{"why":"Shows the $C^0$ estimate fails for $p\\le -1$, marking the lower endpoint of the theorem's range as optimal.","marker":"[57]"},{"why":"Proves uniqueness of even isotropic solutions for $p\\ge -n$ and $q\\le\\min\\{n,n+p\\}$, used to close Theorem 1.5.","marker":"[21]"},{"why":"Introduces the $q$th dual curvature measure and its integral representation, the object whose near-isotropic uniqueness is at issue.","marker":"[49]"},{"why":"Introduces the $L_p$ dual curvature measure and the Monge–Ampère equation (4) that the theorem solves uniquely.","marker":"[72]"}],"fun_headline_variants":["Sharp C0 estimate yields uniqueness for near-isotropic Minkowski","Minkowski uniqueness extended to -1<p<1 with optimal C0 bound","Near-isotropic Lp dual Minkowski: uniqueness for all p in (-1,1)","Even case solved: uniqueness when -1<p<q<min{n,n+p}","Uniqueness near q=n: optimal C0 estimate closes p range"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The weakest link is the proof that any candidate body must stay bounded and cannot collapse to zero volume: that step is a contradiction argument whose punchline is a previously established theorem saying a certain flat limiting configuration cannot exist, and if that theorem does not cover the limit produced by the rescaling, the $C^0$ estimate and the uniqueness conclusion are unsupported.","fun_headline_variants_meta":{"raw":{"variants":["Sharp C0 estimate yields uniqueness for near-isotropic Minkowski","Minkowski uniqueness extended to -1<p<1 with optimal C0 bound","Near-isotropic Lp dual Minkowski: uniqueness for all p in (-1,1)","Even case solved: uniqueness when -1<p<q<min{n,n+p}","Uniqueness near q=n: optimal C0 estimate closes p range"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000158,"raw_usage":{"total_tokens":1252,"prompt_tokens":999,"completion_tokens":253,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":615,"completion_tokens_details":{"reasoning_tokens":148}},"tokens_in":615,"tokens_out":253,"duration_ms":2869,"temperature":1.0,"reasoning_tokens":148,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:28:17.701003+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix $n=2$, $p\\in(-1,0)$, $q=2$, and take $f_\\varepsilon=1+\\varepsilon\\,Y_{1,0}$ on $S^1$, with $Y_{1,0}$ the first spherical harmonic and $\\varepsilon$ small. Theorem 1.3 predicts that $d\\tilde C_{p,2,K}=f_\\varepsilon\\,dH^1$ has exactly one convex solution, with support function $1-\\varepsilon(2-p)^{-1}Y_{1,0}+O(\\varepsilon^2)$; a second solution branch, a bifurcation at $\\varepsilon=0$, or a sequence of bodies with nearly uniform $\\tilde C_{p,2,K}$ whose diameters tend to infinity would refute the theorem.","supporting_citations":[{"cited_title":"Brendle, Kyeongsu Choi, P","cited_arxiv_id":null,"evidence_quote":"Supplies the uniqueness of the exactly isotropic solution, used to identify the limiting body in the compactness argument as the unit ball."},{"cited_title":"B¨ or¨ oczky, C","cited_arxiv_id":null,"evidence_quote":"Establishes near-isotropic uniqueness for $p\\in[0,1)$ and, together with [85], provides the non-existence theorem that rules out degenerate blow-up limits."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the proof pattern for Section 7: local invertibility of the linearized map plus a $C^0$ estimate yields global uniqueness near the constant solution."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows the $C^0$ estimate fails for $p\\le -1$, marking the lower endpoint of the theorem's range as optimal."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves uniqueness of even isotropic solutions for $p\\ge -n$ and $q\\le\\min\\{n,n+p\\}$, used to close Theorem 1.5."},{"cited_title":"Lutwak, Deane Yang, Gaoyong Zhang: Geome tric measures in the dual Brunn-Minkowski theory and their assoc iated Minkowski problems","cited_arxiv_id":null,"evidence_quote":"Introduces the $q$th dual curvature measure and its integral representation, the object whose near-isotropic uniqueness is at issue."}],"review_version":1}