{"id":"59f49d67-0552-40f9-a57c-d29d66258ed5","arxiv_id":"2506.14735","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An α-concave measure with prescribed Euclidean surface area measure exists when the target measure has finite first moment, zero barycenter, and full-dimensional support; the true α-concave function version remains open.","lead":"This paper defines surface area measures for α-concave functions with α between -1/n and 0 and proves an existence result for the associated Minkowski problem via optimal transport. It is a mathematical extension of log-concave moment measure theory to a harder parameter range, though the original function-level problem remains open.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the flagged coercivity step is valid and the measure-level scope is explicitly acknowledged.","rationale":"I read the paper in good faith and focused on Theorem 4.11, the central existence claim. The optimal-transport argument is coherent: Proposition 4.7 characterizes minimizers of (4.5) through a first-order condition, Proposition 4.9 supplies existence via tightness and lower semicontinuity, and Proposition 4.10 establishes essential continuity of the base, which kills the spherical surface area measure. The reader's weakest assumption targets the coercivity inference in Proposition 4.9. That inference is terse but correct: the missing halfspace argument, using barycenter 0 and the non-hyperplane condition on μ, closes the gap. The remaining reader concerns are legitimate but do not undermine Theorem 4.11. Lemma 3.6 is quoted from unpublished work [25] and is needed for the variational formula of Section 3, but it is not used in the optimal-transport construction of Section 4. The function-level Problem 3.14 is not solved unless inf φ>1/α, and the paper states this limitation explicitly. A small presentational gap is that the finiteness of \\barϱ is not checked, but it follows from coercivity and Lemma 2.2. Therefore the conditional verdict remains appropriate, with no need for a stronger rejection.","tokens_in":40419,"tokens_out":54335,"duration_ms":507729,"concrete_test":"Independently re-derive the convex-analytic implication used in Proposition 4.9: for a proper lsc convex φ1 with ∫φ1* dμ<∞, where μ has barycenter 0 and is not supported in any hyperplane, show that non-coercivity would force φ1*=∞ on an open halfspace {⟨y,u⟩>0}, and hence μ would be supported on {⟨y,u⟩=0}, a contradiction. Then apply Lemma 2.2 to the coercive function φ0=φ1+(c0+1)/α to confirm ∫(1-αφ0)^{1/α}dx<∞, so the α-concave measure constructed in Theorem 4.11 is indeed finite and Definition 4.4 applies.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No fatal flaw uncovered in the optimal-transport proof of Theorem 4.11. The fragile bridge identified by the reader—the claim in Proposition 4.9 that an optimal potential φ1 is coercive, since otherwise φ1* would be infinite on a half space—is sound. If a proper lsc convex function φ1 were not coercive, its conjugate would be infinite on an open halfspace {⟨y,u⟩>0}; because ∫φ1* dμ<∞, μ would vanish on that halfspace. With barycenter at the origin, μ would then be forced onto the boundary hyperplane {⟨y,u⟩=0}, contradicting the assumption that μ is not supported in any hyperplane. Hence φ1 is coercive and the subsequent first-moment bound is justified. The genuine caveat is scope: Theorem 4.11 produces an α-concave measure, possibly with a nontrivial singular part on {φ0=1/α}, and does not solve the function-level Problem 3.14 unless inf φ0>1/α, which the paper explicitly leaves open. A minor omission is that the finiteness of \\barϱ=(1-αφ0)^{1/α}dx+ϱ0^s is not verified in the proof, but it follows from the coercivity of φ0 via Lemma 2.2.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops notions of Euclidean and spherical surface area measures for α-concave functions on R^n when -1/n<α<0, via a first-variation formula for the total mass under the α-sum operation, and then extends these notions to α-concave measures. The main variational result is Theorem 3.11, which identifies δJ_α(f,g) as the sum of a bulk integral involving ψ*(∇φ) and a boundary integral over ∂K_f. Section 4 establishes necessary conditions for the functional Minkowski problem (Theorem 4.3) and then, using optimal transport, proves the main existence result Theorem 4.11: for any probability measure μ with finite first moment, barycenter at the origin, and support not contained in a hyperplane, there exists an α-concave measure \\barϱ=(1-αφ_0)^{1/α}dx+ϱ_0^s whose Euclidean surface area measure is μ and whose spherical surface area measure is zero. The authors state explicitly that Problem 3.14 for α-concave functions, rather than measures, remains open.","tokens_in":40622,"tokens_out":12302,"duration_ms":113761,"significance":"If the proofs hold, this is a valuable contribution to the functional Brunn-Minkowski program and to the optimal-transport approach to Minkowski-type problems, extending work of Santambrogio for log-concave functions to the α-concave range with a possibly singular part. The optimal-transport argument in Section 4 is largely self-contained and does not use the target result as an input. I specifically checked the coercivity step in Proposition 4.9: if φ_1 were not coercive, φ_1^* would be +∞ on an open half-space, forcing μ to vanish on that half-space; with barycenter zero this would put μ on a hyperplane, contradicting the assumption. This step is sound. The manuscript is also transparent about its main limitation, namely that Theorem 4.11 solves the extended measure-level problem and leaves the function-level Problem 3.14 open. The principal weakness is that the variational formula in Section 3 relies on Lemma 3.6, whose proof is omitted and imported from an unpublished preprint.","major_comments":[{"comment":"Lemma 3.6 is a load-bearing ingredient for the boundary term in the advertised variational formula (3.43) and hence for Theorem 3.11, but no proof is supplied: the text states that it 'follows from [25, Lemma 5.3] verbatim and is omitted.' Since reference [25] is an arXiv preprint rather than a published source, the derivation of the variational formula is not fully verifiable from the present manuscript. Please reproduce the proof of Lemma 3.6 in this paper, or ensure that the cited result is available in a published reference. This does not affect the optimal-transport part of Section 4, but it is essential for the completeness of Section 3.","section":"Lemma 3.6 and Theorem 3.8"}],"minor_comments":[{"comment":"The statement constructs \\barϱ=(1-αφ_0)^{1/α}dx+ϱ_0^s, but the finiteness of this measure is not verified in the proof. It follows from the coercivity of φ_0 (so that φ_0-1/α≥0 is coercive) together with Lemma 2.2, but this should be stated explicitly.","section":"Theorem 4.11"},{"comment":"The title and abstract refer to a 'Minkowski problem for α-concave functions,' but the main existence theorem is for α-concave measures, and the authors explicitly leave the function-level Problem 3.14 open. The abstract should state this scope limitation clearly to avoid overclaiming.","section":"Abstract and Section 4.2"},{"comment":"There is a typo: 'Furthurmore' should be 'Furthermore.'","section":"Section 2.2, Theorem 2.3"},{"comment":"In the chain of inequalities for T(ϱ_ε,μ), the notation T(ρ_0,μ) is used where T(ϱ_0,μ) is meant, since the transport term includes the singular part; please correct this for notational consistency.","section":"Equation (4.25), Section 4.2"},{"comment":"In the dominated-convergence display after (2.19), the limit variable is written as 'x→∞' where 'n→∞' is intended. Also, the phrase 'lim_{x→∞}' should be 'lim_{n→∞}' throughout that passage.","section":"Proposition 2.5, proof"},{"comment":"There are several typographical errors, including 'summaried' in the introduction and 'Lebesgure' in Section 3; a careful proofreading pass is recommended.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main mathematical claims appear sound, and the optimal-transport proof of Theorem 4.11 is convincing. My main reservation is the incomplete verification of the Section 3 variational formula: Lemma 3.6 is cited from a preprint co-authored by one of the present authors, and its proof is omitted. I would recommend that the editor require the proof of Lemma 3.6 to be included, or at least that the authors confirm the reference has appeared or will appear in a peer-reviewed venue. The Section 4 result is independent of this issue and should survive regardless."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe headline is that this is a solid paper. It generalizes Santambrogio's optimal-transport solution of the log-concave moment measure problem to α-concave functions for α∈(-1/n,0), and in doing so introduces variational formulas for Euclidean and spherical surface area measures in this range. The main theorem, Theorem 4.11, is an existence result at the level of α-concave measures; it is not the solution of the original function-level Problem 3.14, and the paper says so explicitly.\n\nWhat is genuinely new: the definitions of surface area measures for α<0 with the singular-part handling, and the variational formula (Theorem 3.11) that extends Ulivelli's special case under a weaker condition. The optimal transport machinery is used carefully: the Knott–Smith criterion, the coercivity argument in Proposition 4.9, and the lower-semicontinuity of the functional all check out. I did not find a gap in the central derivation.\n\nThe soft spots are real but not fatal. First, the gap between Problem 3.14 and Theorem 4.11 is structural: the constructed solution may have a nontrivial singular part, and the condition inf φ>1/α is left open. That is a genuine boundary of the result, not a flaw, but the abstract and title slightly overstate the scope by saying they study the Minkowski problem for α-concave functions. Second, Lemma 3.6 is quoted verbatim from the unpublished preprint [25] by two of the authors, and it is load-bearing for the boundary term in the variational formula. A reader deserves either a proof or a published pointer. That request is fair in refereeing.\n\nThe necessary conditions in Theorem 4.3 are standard: finite first moment, barycenter at origin, support not in any hyperplane. The sufficiency at the measure level is the actual contribution. This paper will interest convex geometers and functional analysts working on Minkowski-type problems. It deserves a serious referee; I would send it out with a request to address the two points above.","headline":"Solid extension of Santambrogio's moment-measure theorem to α∈(-1/n,0), with the honest caveat that it solves the measure-level Minkowski problem rather than the original function-level one.","tokens_in":41205,"tokens_out":2026,"would_cite":true,"duration_ms":18476,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["26B25","52A40","52A41","35G20","31B99"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that every probability measure with finite first moment, barycenter at the origin, and support not in any hyperplane is the Euclidean surface area measure of some α-concave measure with zero spherical surface area, for…","keywords":["alpha-concave functions","alpha-concave measures","surface area measure","Minkowski problem","optimal transport","first variation","Monge-Ampere equation","moment measures"],"falsifier":"Take $n=1$, $\\alpha=-1/2$, and $\\mu=(\\delta_{-1}+\\delta_1)/2$; this measure has finite first moment, barycenter $0$, and is not supported on a hyperplane. Solve the one-dimensional minimization problem $\\inf\\{(1-\\alpha)F_\\alpha(\\varrho)-\\alpha T(\\varrho,\\mu)\\}$ explicitly and check whether the resulting potential $\\varphi_0$ is coercive and whether the singular part of the minimizer lies in $\\{\\varphi_0=1/\\alpha\\}$. Alternatively, for a non-atomic $\\mu$ in any dimension, test whether the constructed solution has $\\inf\\varphi_0>1/\\alpha$; if it does, the original function-level Minkowski problem is solved for that $\\mu$, and if not, the singular part is genuinely needed.","tokens_in":40191,"feed_emoji":"📐","tokens_out":11253,"duration_ms":95522,"temperature":0.7,"pith_summary":"This paper proves that, for $\\alpha\\in(-1/n,0)$, every probability measure $\\mu$ on $\\mathbb{R}^n$ with finite first moment, barycenter at the origin, and support not lying in any hyperplane is the Euclidean surface area measure of some $\\alpha$-concave measure whose spherical surface area measure is zero. This is the sufficiency half of a functional analogue of the classical Minkowski problem: instead of a convex body, one seeks an $\\alpha$-concave function or measure whose surface-area data recover a prescribed measure. The authors first derive a variational formula for the first variation of total mass under the $\\alpha$-sum, which defines the two surface-area measures, and then solve the existence problem by optimal transport. A sympathetic reader should care because the theorem gives a clean prescription of which measures are surface-area measures in the negative-exponent range, extending the log-concave moment-measure theory to $\\alpha$-concave objects.","feed_headline":"Every balanced measure is an α-concave surface-area measure","feed_subtitle":"Optimal transport builds an α-concave measure whose Euclidean surface area is exactly the given μ.","key_machinery":"The central object is the $\\alpha$-sum $f\\oplus_\\alpha t\\bullet_\\alpha g=(1-\\alpha(\\varphi^*+t\\psi^*)^*)^{1/\\alpha}$, whose first variation turns total mass into a linear functional with integrands that define the Euclidean surface area measure $(\\nabla\\varphi)_\\sharp(f^{1-\\alpha}dx)$ and the spherical surface area measure $(\\nu_{K_f})_\\sharp(f\\,dH^{n-1}|_{\\partial K_f})$. To prove existence, the paper minimizes the functional $(1-\\alpha)F_\\alpha(\\varrho)-\\alpha T(\\varrho,\\mu)$ over probability measures, where $F_\\alpha(\\varrho)=-\\int\\rho^{1/(1-\\alpha)}dx$ and $T$ is the maximal-correlation functional of optimal transport. The minimizer is shown to have the form $(1-\\alpha\\varphi_0)^{1/\\alpha-1}dx+\\varrho_0^s$ with $\\varrho_0^s$ supported on $\\{\\varphi_0=1/\\alpha\\}$, and the Knott-Smith criterion guarantees an optimal plan whose support lies in $\\mathrm{Graph}(\\partial\\varphi_0)$, which is precisely what Definition 4.4 requires for $\\mu$ to be the Euclidean surface area measure.","core_discovery":"Let $-1/n<\\alpha<0$ and let $\\mu$ be a probability measure with finite first moment, barycenter at the origin, and support not contained in any hyperplane. The paper establishes that there exists an $\\alpha$-concave measure $\\bar\\varrho=(1-\\alpha\\varphi_0)^{1/\\alpha}\\,dx+\\varrho_0^s$ such that the Euclidean surface area measure of $\\bar\\varrho$ is exactly $\\mu$ and its spherical surface area measure is the zero measure. The proof begins with the first-variation identity $\\delta J_\\alpha(f,g)=\\int_{\\mathbb{R}^n}\\psi^*(\\nabla\\varphi)f^{1-\\alpha}dx+\\int_{\\partial K_f}h_{D\\psi}(\\nu)f\\,dH^{n-1}$, which gives the pair of surface-area measures. These notions are then extended to $\\alpha$-concave measures, allowing a singular part supported on $\\{\\varphi_0=1/\\alpha\\}$, and the existence theorem is obtained by minimizing $(1-\\alpha)F_\\alpha(\\varrho)-\\alpha T(\\varrho,\\mu)$ and applying the Knott-Smith optimality criterion to place the optimal plan inside the graph of $\\partial\\varphi_0$. The same three conditions on $\\mu$ had already been shown necessary for Euclidean surface-area measures of $\\alpha$-concave functions with essentially continuous base, so the theorem completes the characterization at the level of $\\alpha$-concave measures.","pith_inferences":["If the coercivity inference in Proposition 4.9 could be replaced by a weaker growth hypothesis, the same optimal-transport strategy would likely cover exponents outside $(-1/n,0)$ or Orlicz-type functionals where finite first moment alone does not force superlinear potentials.","The authors' open question about $\\inf\\varphi_0>1/\\alpha$ suggests a testable dichotomy: for sufficiently regular, fast-decaying $\\mu$ the singular part should vanish, so one could predict exactly when the function-level problem has a solution.","Because the singular part is concentrated on $\\{\\varphi_0=1/\\alpha\\}$, the construction shows how point masses in $\\mu$ are absorbed by 'infinite-density' regions of the $\\alpha$-concave measure; this mechanism may transfer to other functional Brunn–Minkowski problems.","One could numerically solve the one-dimensional minimization for representative measures to map the boundary between the regular regime ($\\inf\\varphi_0>1/\\alpha$) and the singular regime, giving explicit evidence for when Problem 3.14 is solvable."],"forward_implications":["For every $\\mu$ satisfying finite first moment, barycenter at the origin, and non-planar support, Theorem 4.11 produces an $\\alpha$-concave measure with Euclidean surface area measure $\\mu$ and zero spherical surface area measure; this is the sufficiency half of the characterization begun in Theorem 4.3.","Whenever the constructed potential has $\\inf\\varphi_0>1/\\alpha$, the solution is an ordinary $\\alpha$-concave function and the original Euclidean functional Minkowski problem (Problem 3.14) is solved; the authors leave open when this happens.","In the smooth case the extended problem is equivalent to the Monge–Ampère equation $h(\\nabla\\varphi)\\det(\\nabla^2\\varphi)=(1-\\alpha\\varphi)^{(1-\\alpha)/\\alpha}$, so the theorem supplies a weak, measure-valued solution for arbitrary measures in the stated class.","The result interpolates the classical convex-body Minkowski theorem and the log-concave moment-measure theorem: as $\\alpha$ moves through $(-1/n,0)$, characteristic functions of convex bodies and log-concave densities sit at the two ends.","Together with the necessity results, the theorem says the triple of conditions—finite first moment, centered barycenter, and non-degeneracy—is exactly what characterizes Euclidean surface-area measures of $\\alpha$-concave measures."],"supporting_citations":[{"why":"Supplies the Knott-Smith optimality criterion: an optimal plan for the maximal-correlation functional is supported on the subgradient graph, the bridge from the variational problem to Definition 4.4.","marker":"[40]"},{"why":"Introduces the optimal-transport strategy for moment measures by minimizing an entropy-type functional minus the maximal-correlation term, the template for the proof of Theorem 4.11.","marker":"[56]"},{"why":"Extends the moment-measure method to q-moment measures and shows how to handle singular parts, closely followed in Section 4.2.","marker":"[37]"},{"why":"Provides the necessary-condition lemmas for moment measures (finite first moment, zero barycenter, non-planar support) and the essential-continuity criterion used in Theorem 4.3 and Proposition 4.10.","marker":"[21]"},{"why":"Provides the radial-function variational lemma and generalized dominated-convergence framework used to prove the first-variation formula in Theorem 3.10.","marker":"[25]"},{"why":"Defines the $\\alpha$-sum and $\\alpha$-scalar multiplication on $\\alpha$-concave functions, the operation whose first variation defines the surface-area measures.","marker":"[47]"},{"why":"Supplies the convex-body dual-volume variational formula that is extended to unbounded closed convex sets in the proof of the functional variation.","marker":"[34]"},{"why":"Establishes support functions and mean width for $\\alpha$-concave functions, the language used to state the geometry of the $\\alpha$-sum.","marker":"[51]"}],"fun_headline_variants":["α-concave surface-area measures exist for every balanced μ","Optimal transport proves Minkowski for α-concave measures","Balanced measures realized as α-concave Euclidean surface areas","Minkowski problem for α-concave measures settled via OT","From any balanced μ, an α-concave measure with that surface area"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing step is the inference in Proposition 4.9 that the optimal-transport dual potential $\\varphi_1$ is coercive whenever $\\mu$ has finite first moment; if that inference fails for some measure, the minimizer need not have the growth that places the singular part on $\\{\\varphi_0=1/\\alpha\\}$, and the representation in Definition 4.4 would not apply.","fun_headline_variants_meta":{"raw":{"variants":["α-concave surface-area measures exist for every balanced μ","Optimal transport proves Minkowski for α-concave measures","Balanced measures realized as α-concave Euclidean surface areas","Minkowski problem for α-concave measures settled via OT","From any balanced μ, an α-concave measure with that surface area"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000261,"raw_usage":{"total_tokens":1583,"prompt_tokens":923,"completion_tokens":660,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":539,"completion_tokens_details":{"reasoning_tokens":569}},"tokens_in":539,"tokens_out":660,"duration_ms":6675,"temperature":1.0,"reasoning_tokens":569,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:48:10.121357+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $n=1$, $\\alpha=-1/2$, and $\\mu=(\\delta_{-1}+\\delta_1)/2$; this measure has finite first moment, barycenter $0$, and is not supported on a hyperplane. Solve the one-dimensional minimization problem $\\inf\\{(1-\\alpha)F_\\alpha(\\varrho)-\\alpha T(\\varrho,\\mu)\\}$ explicitly and check whether the resulting potential $\\varphi_0$ is coercive and whether the singular part of the minimizer lies in $\\{\\varphi_0=1/\\alpha\\}$. Alternatively, for a non-atomic $\\mu$ in any dimension, test whether the constructed solution has $\\inf\\varphi_0>1/\\alpha$; if it does, the original function-level Minkowski problem is solved for that $\\mu$, and if not, the singular part is genuinely needed.","supporting_citations":[{"cited_title":"S.: On the optimal mapping of distributions","cited_arxiv_id":null,"evidence_quote":"Supplies the Knott-Smith optimality criterion: an optimal plan for the maximal-correlation functional is supported on the subgradient graph, the bridge from the variational problem to Definition 4.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the optimal-transport strategy for moment measures by minimizing an entropy-type functional minus the maximal-correlation term, the template for the proof of Theorem 4.11."},{"cited_title":"Journal of Convex Analysis","cited_arxiv_id":null,"evidence_quote":"Extends the moment-measure method to q-moment measures and shows how to handle singular parts, closely followed in Section 4.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the necessary-condition lemmas for moment measures (finite first moment, zero barycenter, non-planar support) and the essential-continuity criterion used in Theorem 4.3 and Proposition 4.10."},{"cited_title":"Electron","cited_arxiv_id":null,"evidence_quote":"Defines the $\\alpha$-sum and $\\alpha$-scalar multiplication on $\\alpha$-concave functions, the operation whose first variation defines the surface-area measures."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the convex-body dual-volume variational formula that is extended to unbounded closed convex sets in the proof of the functional variation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes support functions and mean width for $\\alpha$-concave functions, the language used to state the geometry of the $\\alpha$-sum."}],"review_version":2}