{"id":"164bdc0f-ad16-42a6-8cdc-8e811734303c","arxiv_id":"2608.01912","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"In the plane, for 0<q<2, equal dual curvature measures force origin-symmetric convex bodies to coincide; for q>n in R^n, distinct origin-symmetric bodies can share the same dual curvature measure.","lead":"This mathematics paper determines exactly when two symmetric convex shapes that have the same dual curvature data must be the same shape, and when they can differ. It resolves a long-open uniqueness question for the plane and extends the known range of non-uniqueness to all dimensions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 1.1 depends on the equality characterization of the planar log-Brunn–Minkowski inequality from [5]; this external equality input is the least secure step, but the all-scales argument in Lemma 3.3 likely neutralizes the main exceptional case.","rationale":"The paper's central results are internally consistent. I checked the main computational steps: the Prékopa–Leindler transfer in Lemma 3.3, the concavity argument in Theorem 1.1 Step 1, the second-derivative computation in Theorem 1.4, the parallelotope construction for n≥3, and the variational reduction in Theorem 1.2. All appear correct. The weakest assumption is indeed the external equality characterization from [5], as the reader noted. This is a legitimate load-bearing concern because without the equality cases, the uniqueness proof fails. However, the cited theorem is published and the paper's use of it is precise: Lemma 3.3 relies on the direct-sum equality condition, and the simultaneous-truncation argument would rule out the standard non-dilate parallelogram equality examples, which only satisfy volume equality for the full bodies, not for all truncated pairs. Therefore the concern does not rise to the level of a demonstrated flaw. The verdict should remain ACCEPT, with the caveat that independent verification of the [5] equality cases would strengthen confidence. No ad hominem or theatrical language is intended; this is a straightforward external-dependency check.","tokens_in":14117,"tokens_out":38489,"duration_ms":320815,"concrete_test":"Independently verify the equality-case theorem in Böröczky–Lutwak–Yang–Zhang [5] (or a subsequent authoritative source) for the planar log-BM inequality: (i) confirm that equality in (3.1) implies the direct-sum decomposition with homothetic components, and (ii) confirm that when one body has no non-trivial direct-sum decomposition (e.g., K∩rB^n for r∈(r_-(K), r_+(K)) by Lemma 3.2), equality forces the two bodies to be dilates. If [5] only proves the inequality without equality cases, or if a non-dilate parallelogram pair can satisfy equality for all truncations in Lemma 3.3, then Theorem 1.3's equality condition and Theorem 1.1 would need an alternative proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 relies on Theorem 1.3, whose equality condition is obtained in Lemma 3.3 by transferring equality from the volume log-Brunn–Minkowski inequality (3.1) to the dual inequality (3.2). Lemma 3.3 assumes the full direct-sum equality characterization of the log-BM conjecture as stated in [5]. If [5] proved only the inequality, or if its equality cases include non-dilate configurations that are not ruled out for the specific truncated bodies K∩rB^n and L∩e^τ rB^n, then the conclusion that equality in (3.2) forces K and L to be dilates is unsupported. The usual parallelogram-type equality cases (m=2 in the direct-sum condition) are the potential worry. However, Lemma 3.3 obtains equality in (3.1) for every truncation simultaneously; this multi-scale structure would force the homothety ratio to be constant across scales and thereby exclude parallelogram-type exceptional cases that hold only for the full bodies. Thus the concern is real but likely benign: the external equality characterization is load-bearing, but the paper's argument uses it in a way that appears compatible with the known planar equality cases.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies uniqueness and non-uniqueness for the even dual Minkowski problem associated with the dual curvature measures \\tilde C_q. The main results are: in \\mathbb R^2, for 0<q<2, origin-symmetric convex bodies with equal dual curvature measures must coincide (Theorem 1.1); and for q>n\\ge 2, there exist distinct origin-symmetric bodies with the same dual curvature measure (Theorem 1.2), with the construction upgraded to smooth bodies in Remark 4.2. These results are obtained by establishing a logarithmic Brunn–Minkowski inequality for dual quermassintegrals in the plane with equality characterization (Theorem 1.3), and by constructing explicit parallelotope counterexamples showing that the corresponding inequality fails for q>n (Theorem 1.4). The proof transfers the planar log-Brunn–Minkowski inequality to dual quermassintegrals through a Prékopa–Leindler layer-cake argument, uses concavity and variational arguments for the uniqueness part, and combines explicit rectangle counterexamples with a compactness argument for the non-uniqueness part.","tokens_in":14312,"tokens_out":62050,"duration_ms":499267,"significance":"If the results are correct, the paper substantially sharpens the known non-uniqueness range for the even dual Minkowski problem from q>2n to the borderline q>n, and it provides the first global uniqueness statement for positive q in the plane, for 0<q<2. The arguments are well-structured and mostly self-contained: the layer-cake transfer in Lemma 3.3 is elegant, the equality analysis uses a multi-scale truncation argument that appears to rule out the parallelogram exceptional cases of the classical planar log-Brunn–Minkowski inequality, and the counterexample in Theorem 1.4 is explicit and computable. The paper also gives a concrete failure of the conjectured dual log-Brunn–Minkowski inequality, which is a falsifiable statement in its own right. The exposition is clear, and the main computations in Sections 3 and 4 are reproducible.","major_comments":[],"minor_comments":[{"comment":"The proof of Theorem 1.3 relies on the equality characterization of the planar log-Brunn–Minkowski inequality from [5], but Lemma 3.3 is stated conditionally on the direct-sum equality condition and the proof of Theorem 1.3 cites [5] only for the inequality. Please state explicitly the theorem from [5] that supplies the equality cases used here, or add a short proof of the needed equality cases for the truncated bodies K\\cap rB^n and L\\cap e^\\tau rB^n; this is a load-bearing input for Theorem 1.1.","section":"Section 3, Lemma 3.3 and Theorem 1.3"},{"comment":"The notation lin(K_i-K_i) is used without definition; please explain that it denotes the linear subspace parallel to the affine hull of the set K_i-K_i.","section":"Section 2, Lemma 2.2"},{"comment":"The phrase 'Lets s→∞ in the volume identity' contains a typo and should read 'Letting s→∞ in the volume identity'.","section":"Section 3, Lemma 3.3, Step 2"},{"comment":"The sentence 'Since the equality holds at all coordinate directions ±e_i, i=1,...,n, every point of [(h_{K_{t,\\varepsilon}}h_{K_{-t,\\varepsilon}})^{1/2}] lies in S_{t,\\varepsilon}' is terse; it would be clearer to note explicitly that the defining inequalities at v=±e_i give |x_i|\\le r_t for i=1,2 and |x_i|\\le \\varepsilon for i=3,\\ldots,n.","section":"Section 4, Theorem 1.4, Step 2"},{"comment":"The phrase 'our established logarithmic Brunn-Minkowski inequality for dual quermassintegrals' should be rephrased, for example as 'the logarithmic Brunn-Minkowski inequality for dual quermassintegrals established in this paper,' to avoid ambiguity about what is new.","section":"Abstract and Introduction"},{"comment":"Please change 'has different classical solutions' to 'has two different classical solutions' to match the statement that two distinct bodies are produced.","section":"Remark 4.2"}],"recommendation":"minor_revision","confidential_remarks":"The main things to verify before acceptance are the equality characterization of the planar log-Brunn–Minkowski inequality from [5] and its precise statement; the rest of the proof appears sound. If the authors can quote the exact theorem from [5] or otherwise justify the equality input, I would support publication. The paper is a strong contribution to the dual Minkowski problem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Ge and Yang settle the planar uniqueness question for the even dual Minkowski problem in the range 0<q<2, and extend non-uniqueness from q>2n down to q>n in all dimensions. The core new input is a log-Brunn–Minkowski inequality for dual quermassintegrals eV_q in the plane, with equality iff the bodies are dilates, plus a family of rectangles and parallelotopes that violates the inequality for q>n. The proofs are thorough; I checked the main computations (the Prékopa–Leindler transfer, the truncation argument, the second derivative of eV_q(A_t), and the epsilon expansion for parallelotopes) and they are sound.\n\nThe soft spot is the load-bearing external input: the equality characterization of the planar log-Brunn–Minkowski inequality from [5]. Lemma 3.3 converts equality in the dual inequality into equality in the volume log-BM for all truncations K∩rB^n and L∩e^τ rB^n. Lemma 3.2 shows these truncations have no non-trivial direct-sum decomposition for r between inradius and circumradius, which rules out the parallelogram equality cases. So even if [5]'s equality statement includes parallelograms, the multi-scale argument still forces the truncations to be dilates. The reasoning holds up. The only caveat is that [5]'s equality statement is used as a black box, but it is a published theorem and the use is faithful.\n\nThe paper is honest: no fitting, no self-citations, no hidden circularity. Theorem 1.2's contradiction argument is standard and correct. The regularity upgrade via [2] is fine. I see no red flags.\n\nThis deserves a serious referee. It is a within-field advance, not a wide-scope breakthrough, but for convex geometry and spherical Monge–Ampère equations it is a solid step. I would bring it to a reading group and cite it if I worked on dual Minkowski problems. Recommendation: send to peer review.","headline":"Settles planar even dual Minkowski uniqueness for 0<q<2 and extends non-uniqueness to q>n; the proofs are careful and the external equality input checks out.","tokens_in":14929,"tokens_out":27240,"would_cite":true,"duration_ms":202581,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A38","35J96"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the even dual Minkowski problem has a unique origin-symmetric planar solution for 0<q<2, but admits two different smooth solutions for every q>n≥2.","keywords":["convex body","dual Minkowski problem","dual curvature measure","dual quermassintegral","logarithmic Brunn-Minkowski inequality","Monge-Ampère equation","uniqueness","non-uniqueness"],"falsifier":"At $q=1$, solve the even dual Minkowski problem for an arbitrary smooth positive even density $f$: find two distinct origin-symmetric smooth convex bodies $K,L$ satisfying $\\tilde C_1(K,\\cdot)=\\tilde C_1(L,\\cdot)=f\\,du$. The paper's Theorem 1.1 says no such pair exists; producing one would refute the uniqueness claim.","tokens_in":13861,"feed_emoji":"📐","tokens_out":14521,"duration_ms":117731,"temperature":0.7,"pith_summary":"This paper settles two ends of the uniqueness question for the even dual Minkowski problem, which asks whether a convex body is determined by its dual curvature measures. The authors prove that in the plane, for exponents $0<q<2$, equal dual curvature measures force two origin-symmetric bodies to be identical. They also prove that for $q>n$ in every dimension $n\\ge 2$, uniqueness fails: different origin-symmetric bodies can carry the same dual curvature measure, and the two bodies can be chosen smooth, so the associated spherical Monge–Ampère equation has at least two classical solutions. Both results are governed by a logarithmic Brunn–Minkowski inequality for dual quermassintegrals that the paper establishes in the plane and shows to fail for $q>n$.","feed_headline":"For q>n, one dual curvature measure yields two symmetric bodies","feed_subtitle":"For 0<q<2 in the plane the same data force one body; for q>n at least two smooth solutions exist.","key_machinery":"The central object is the $q$-th dual curvature measure $\\tilde C_q(K,\\cdot)$ and its total mass, the $q$-th dual quermassintegral $\\tilde V_q(K)=\\frac1n\\int_{S^{n-1}}\\rho_K(u)^q\\,du$, where $\\rho_K$ is the radial function. The mechanism is the logarithmic Brunn–Minkowski inequality for these integrals, $\\tilde V_q((1-\\lambda)\\cdot K+_0\\lambda\\cdot L)\\ge \\tilde V_q(K)^{1-\\lambda}\\tilde V_q(L)^\\lambda$, with the Wulff shape $(1-\\lambda)\\cdot K+_0\\lambda\\cdot L=[h_K^{1-\\lambda}h_L^\\lambda]$ as the interpolation. The paper derives the planar inequality for $0<q<2$ by a one-dimensional Prékopa–Leindler step applied to the layer-cake formula $\\tilde V_q(A)=\\frac{q(n-q)}{n}\\int_0^\\infty r^{q-n-1}V_n(A\\cap rB^n)\\,dr$, and extracts equality from the equality cases of the volume logarithmic Brunn–Minkowski inequality. For the non-uniqueness range $q>n$, the same Wulff-shape machinery is used in reverse: an explicit family of rectangles $A_t=[-(1+t),1+t]\\times[-(1-t),1-t]$ satisfies $\\tilde V_q(A_t)=\\tilde V_q(A_{-t})$ while $\\tilde V_q([(h_{A_t}h_{A_{-t}})^{1/2}])$ is strictly smaller than the geometric mean, and taking Cartesian products with small cubes lifts the strict failure to $\\mathbb{R}^n$.","core_discovery":"The paper's central discovery is a sharp transition in the exponent $q$ for the even dual Minkowski problem. For $n=2$ and $0<q<2$, the problem is well-posed among origin-symmetric convex bodies: $\\tilde C_q(K,\\cdot)=\\tilde C_q(L,\\cdot)$ implies $K=L$. For $q>n\\ge 2$, it is not: there exist origin-symmetric convex bodies $K,L$, smooth and uniformly convex, with $\\tilde C_q(K,\\cdot)=\\tilde C_q(L,\\cdot)$ but $K\\neq L$. Equivalently, the spherical Monge–Ampère equation $h(u)|\\nabla h(u)|^{q-n}\\det(h_{ij}(u)+h(u)\\delta_{ij})=f$ admits two different classical solutions for a suitable smooth positive even density $f$. The proof is driven by the logarithmic Brunn–Minkowski inequality for the dual quermassintegrals $\\tilde V_q$: the paper proves it in $\\mathbb{R}^2$ for $0<q<2$ with equality only for dilates, and disproves it for $q>n$ using an explicit family of rectangles, lifted to higher dimensions.","pith_inferences":["If the volume logarithmic Brunn–Minkowski inequality with its conjectured equality cases is eventually proved in higher dimensions, the truncation argument in Lemma 3.3 would extend the planar uniqueness conclusion for $0<q<n$ to those dimensions; the present paper only establishes the planar case.","The boundary exponent $q=n$ is natural: for $n=2$ and $q=2$ the equality theory already gives uniqueness, and the local expansion in Section 4 changes sign exactly when $q$ passes $n$, leaving $q=n$ as the delicate threshold the paper does not address.","The non-uniqueness is probably not confined to the one-parameter rectangle family; any pair of origin-symmetric bodies whose Wulff combination has dual quermassintegral smaller than the geometric mean and whose radial integrals are matched by the $t\\mapsto -t$ symmetry would produce another pair, so examples may be abundant."],"forward_implications":["For $n=2$ and $0<q<2$, the even dual Minkowski problem is unique among origin-symmetric bodies: the data determine the body.","For every $q>n\\ge 2$, the even dual Minkowski problem admits at least two distinct smooth origin-symmetric solutions with the same data, so well-posedness fails in this range.","The spherical Monge–Ampère equation $h(u)|\\nabla h(u)|^{q-n}\\det(h_{ij}(u)+h(u)\\delta_{ij})=f$ has a smooth positive even density $f$ with at least two classical solutions for each $q>n$.","The explicit rectangle construction shows the logarithmic Brunn–Minkowski inequality for dual quermassintegrals is genuinely false for $q>n$, not merely unproved, so any uniqueness argument in this range must bypass that inequality."],"supporting_citations":[{"why":"Supplies the planar logarithmic Brunn–Minkowski inequality and its equality characterization, the external input behind Theorems 1.1 and 1.3.","marker":"[5]"},{"why":"Introduces dual curvature measures and the dual Minkowski problem, and provides the Wulff-shape variational formula used in both uniqueness and non-uniqueness proofs.","marker":"[21]"},{"why":"Provides the spherical-image/direct-sum lemma used to show truncated bodies admit no nontrivial direct-sum decomposition in the equality step.","marker":"[4]"},{"why":"Regularity theorem used to upgrade the non-unique pairs to smooth $C^\\infty_+$ bodies in Remark 4.2.","marker":"[2]"},{"why":"Earlier planar non-uniqueness for even integer $q\\ge 6$, which the present theorem extends to all $q>2$ when $n=2$.","marker":"[20]"},{"why":"Earlier non-uniqueness for $q>2n$ in all dimensions, which this paper improves to $q>n$.","marker":"[9]"}],"fun_headline_variants":["Even dual Minkowski: q>n yields many, q<2 in plane unique","One dual curvature measure can force two smooth bodies for q>n","Log-Brunn-Minkowski splits dual Minkowski into unique and non-unique","Dual curvature measures: unique in plane, multiple for q>n"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The planar uniqueness result rests on an external theorem that characterizes exactly when equality holds in the logarithmic Brunn–Minkowski inequality; if that characterization has hidden exceptional cases, the conclusion that equal dual curvature measures force the bodies to be identical could fail.","fun_headline_variants_meta":{"raw":{"variants":["Even dual Minkowski: q>n yields many, q<2 in plane unique","One dual curvature measure can force two smooth bodies for q>n","Log-Brunn-Minkowski splits dual Minkowski into unique and non-unique","Dual curvature measures: unique in plane, multiple for q>n"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001006,"raw_usage":{"total_tokens":4218,"prompt_tokens":875,"completion_tokens":3343,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":491,"completion_tokens_details":{"reasoning_tokens":3257}},"tokens_in":491,"tokens_out":3343,"duration_ms":23031,"temperature":1.0,"reasoning_tokens":3257,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:05:49.713652+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"At $q=1$, solve the even dual Minkowski problem for an arbitrary smooth positive even density $f$: find two distinct origin-symmetric smooth convex bodies $K,L$ satisfying $\\tilde C_1(K,\\cdot)=\\tilde C_1(L,\\cdot)=f\\,du$. The paper's Theorem 1.1 says no such pair exists; producing one would refute the uniqueness claim.","supporting_citations":[{"cited_title":"B¨ or¨ oczky, E","cited_arxiv_id":null,"evidence_quote":"Supplies the planar logarithmic Brunn–Minkowski inequality and its equality characterization, the external input behind Theorems 1.1 and 1.3."},{"cited_title":"Huang, E","cited_arxiv_id":null,"evidence_quote":"Introduces dual curvature measures and the dual Minkowski problem, and provides the Wulff-shape variational formula used in both uniqueness and non-uniqueness proofs."},{"cited_title":"B¨ or¨ oczky, P","cited_arxiv_id":null,"evidence_quote":"Provides the spherical-image/direct-sum lemma used to show truncated bodies admit no nontrivial direct-sum decomposition in the equality step."},{"cited_title":"B¨ or¨ oczky, F","cited_arxiv_id":null,"evidence_quote":"Regularity theorem used to upgrade the non-unique pairs to smooth $C^\\infty_+$ bodies in Remark 4.2."},{"cited_title":"Huang, Y","cited_arxiv_id":null,"evidence_quote":"Earlier planar non-uniqueness for even integer $q\\ge 6$, which the present theorem extends to all $q>2$ when $n=2$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier non-uniqueness for $q>2n$ in all dimensions, which this paper improves to $q>n$."}],"review_version":2}