{"id":"2ac9b3a9-0dc3-4dfc-827e-e932044d69f0","arxiv_id":"2508.12028","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The Gaussian Minkowski problem is generalized to epigraphs of convex functions, and existence of convex functions realizing prescribed Gaussian moment measures is established under mild conditions.","lead":"This paper introduces a Gaussian moment measure for epigraphs of convex functions and solves the associated Minkowski-type problem, showing that under mild conditions every suitable measure arises from some convex function. It matters because it extends a central problem in convex geometry from convex bodies to unbounded convex graphs, opening a new direction for Gaussian analysis and functional inequalities.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The existence claim rests on the unverified variational formula for Gaussian epigraph volume under infimal convolution; the corrupted text makes this formula uncheckable, and a failure there would collapse the main theorem.","rationale":"The reader's weakest assumption already points at the variational formula, and I agree that it is the load-bearing step. My concern sharpens this: it is not merely that differentiability or representation might fail, but that the unbounded epigraph setting makes the first variation under infimal convolution a non-trivial analytic object, and the corrupted text provides no way to check the hypotheses or the proof. Because the concern is unconfirmed rather than demonstrated, I do not move the verdict: the reader's UNVERDICTED remains appropriate. The proposed concrete test—independent re-derivation plus a direct computation on smooth and nonsmooth examples—would settle whether the variational formula and the measure representation actually hold for the class of functions used in the theorem.","tokens_in":7722,"tokens_out":7390,"duration_ms":82543,"concrete_test":"Obtain the clean arXiv source for 2508.12028 and re-derive the variational formula from the definitions, without relying on the paper's own derivation. Then perform a direct numerical check: take φ(x)=|x|^2/2 in R^2, perturb it by φ_t = φ □ tψ with ψ(x)=v·x for a test vector v, compute the Gaussian volume of epi(φ_t) explicitly or by high-accuracy quadrature, and verify that the t-coefficient equals the claimed moment measure applied to a chosen Borel set. Repeat for the nonsmooth φ(x)=|x|; if no t-derivative exists or the derivative is not a Borel measure, the admissible class must be restricted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—given a Borel measure satisfying mild conditions, there exists a convex function whose Euclidean Gaussian moment measure equals it—depends entirely on 'a variational formula' derived by combining Gaussian volume of epigraph(φ) with the infimal-convolution perturbation. That formula has to yield a first variation representable by a Borel measure on R^n (and a companion measure on S^{n-1}) on the whole class of convex functions admitted in the theorem. If the derivative exists only for smooth or superlinear φ, or if the variation is not a finite measure under the stated 'mild and natural conditions,' the existence proof has no foundation. The supplied full text is corrupted to the point that no equation or proof can be inspected, so I cannot locate the exact assumptions or check the derivation. This is not an accusation of error; it is an identification of the unique point on which the theorem hinges and that currently cannot be verified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript, as supplied, is largely unreadable: only the abstract and scattered fragments of the full text are intelligible. The abstract announces a variational formula obtained by perturbing a convex function φ through infimal convolution, and uses it to define a Borel measure on R^n (called the Euclidean Gaussian moment measure of φ) together with a Borel measure on the unit sphere S^{n-1}. It then claims that the associated Minkowski-type problem is solved under mild and natural conditions on the prescribed measure. No theorem statement, proof, or equation can be inspected in the provided full text.","tokens_in":7902,"tokens_out":2261,"duration_ms":27321,"significance":"If the announced results are correct, the paper would introduce a new Gaussian-type Minkowski problem for epigraphs of convex functions and provide an existence theorem, extending the family of Minkowski problems to a natural class of non-compact convex sets. The variational approach via infimal convolution is a plausible and potentially valuable technique. From the abstract, there is no apparent circularity or free-parameter fitting: the solution is stated to be an existence result for a prescribed measure. However, because the full text is corrupted and no derivation or theorem statement is checkable, the significance cannot be independently verified at this time.","major_comments":[{"comment":"The central existence claim rests entirely on the asserted variational formula obtained by combining the Gaussian volume of epigraph(φ) with the infimal-convolution perturbation of φ. The abstract states neither the formula itself nor the hypotheses under which it holds. In particular, it is not specified which class of convex functions is allowed, whether the Gaussian volume of the epigraph is finite, and in what sense the first variation is represented by a Borel measure on R^n and a companion Borel measure on S^{n-1}. If the first variation fails to be a finite Borel measure under the 'mild and natural conditions' on the prescribed measure, the existence proof would collapse. A precise statement and proof of this variational formula, with explicit hypotheses, is indispensable.","section":"Abstract (variational formula)"},{"comment":"The provided full text is corrupted and cannot be read as mathematics. For example, a line reading 'arXiv:2508.12023v2  [cs.CV]  15 Sep 2025' appears inside the manuscript, and displayed formulas such as those following 'In ? ? ?' and near 'Lemma' are uninterpretable mojibake. No theorem, lemma, or proof can be checked. This is a load-bearing obstruction because the paper's central claim is an existence theorem whose proof depends on the unverifiable variational formula. A clean, complete manuscript is required before any substantive evaluation of correctness is possible.","section":"Full text (all displayed mathematics)"},{"comment":"The abstract says 'the newly posed Minkowski problem is solved' but does not state the problem precisely. It is not visible what data are prescribed (the full measure on R^n? the pair of measures on R^n and S^{n-1}?), what normalization or growth conditions are imposed, and in what class of convex functions (e.g., finite-valued, coercive, or C^1) the solution is sought. Without a precise statement of the main theorem, the existence claim cannot be assessed, and the novelty relative to known Gaussian Minkowski problems cannot be located.","section":"Abstract (problem statement)"}],"minor_comments":[{"comment":"The term 'Euclidean Gaussian moment measure' is introduced without explanation of how it relates to the existing Gaussian Minkowski problem for convex bodies; the paper should include at least one sentence positioning the new measure in that literature.","section":"Abstract (terminology)"},{"comment":"No references are visible in the readable portions; if the full text contains a bibliography, it must be checked for coverage of the recent Gaussian Minkowski problem literature.","section":"General"}],"recommendation":"uncertain","confidential_remarks":"The supplied text appears to be a corrupt PDF-to-text extraction rather than the actual manuscript, with an unrelated arXiv identifier embedded in the body. The reader's low confidence and the skeptic's concern about the variational formula are both warranted. I would recommend requesting a clean, complete version of the paper from the authors before any further review; the current submission cannot be evaluated on its merits."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this looks like a legitimate new Minkowski-type problem, not a rebranding. The Euclidean Gaussian moment measure for epigraphs is a natural object, and solving the associated Minkowski problem under mild assumptions would be a solid extension of the recent Gaussian Minkowski literature. If the proof is right, it's a useful contribution for people working in convex geometry and geometric measure theory.\n\nWhat the paper does well is visible even from the abstract: it identifies the right variational structure (infimal convolution perturbation of the epigraph's Gaussian volume) and uses it to produce a Borel measure on R^n and one on S^{n-1}. That is the standard architecture for these problems. The claim that existence holds under 'mild and natural conditions' on the prescribed measure is strong but plausible.\n\nThe soft spot is entirely practical: the full text I received is unreadable. It's mojibake. No equation, no proof, no theorem statement survives. So I cannot check the variational formula, which is the load-bearing step. Your stress-test note names exactly the right hinge: if the first variation of Gaussian volume under infimal convolution fails to be a finite Borel measure on the relevant class of convex functions, the existence theorem falls. But I have no evidence that it fails; I also have no evidence that it works. The abstract alone gives no details.\n\nI should be clear that this is not a critique of the mathematics. It is a statement about the reviewing environment. I don't want to desk-reject a plausible result just because my copy is corrupted. Nor do I want to accept a claim I can't see. The right move is to get a clean text, then send it to a referee familiar with the Gaussian Minkowski program, who could quickly assess the variational formula and the conditions. If the proof is correct, the paper deserves publication. If there is a gap, it will likely be in the regularity/differentiability of the volume functional.\n\nSo my bottom line: this deserves a serious referee, not a desk reject. But someone needs to obtain a readable version before that can happen. For me personally, I wouldn't cite it yet, and I wouldn't put it on the reading group until I can read it.","headline":"The abstract announces a genuine extension of the Gaussian Minkowski program to epigraphs; the supplied body is too corrupted to verify any of the actual math.","tokens_in":8354,"tokens_out":2673,"would_cite":false,"duration_ms":27113,"reading_group":"maybe","serious_thinker":"unclear","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A41","52A38"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper's central claim is that every Borel measure on $\\mathbb{R}^n$ satisfying mild natural conditions is the Euclidean Gaussian moment measure of the epigraph of some convex function.","keywords":["Gaussian Minkowski problem","convex functions","epigraph","Gaussian volume","infimal convolution","moment measure","variational formula","Borel measure"],"falsifier":"Take a specific finite Borel measure on $\\mathbb{R}^n$ that satisfies the paper's stated conditions but is concentrated on a set where no convex epigraph's Gaussian moment measure could concentrate, and check whether the paper's variational construction still yields a solution; a counterexample would refute the sufficiency claim. A more direct check is to compute the Gaussian volume derivative for an explicit one-dimensional convex function, such as $\\varphi(x)=x^2$, under the infimal-convolution perturbation, and verify that the derivative equals the proposed Euclidean Gaussian moment measure evaluated on the perturbation direction.","tokens_in":7559,"feed_emoji":"📐","tokens_out":5962,"duration_ms":62138,"temperature":0.7,"pith_summary":"The paper studies a Gaussian analogue of the classical Minkowski problem, but for epigraphs of convex functions rather than compact convex bodies. It derives a variational formula: perturbing a convex function $\\varphi$ through infimal convolution changes the Gaussian volume of its epigraph in a way controlled by a Borel measure on $\\mathbb{R}^n$ and a companion Borel measure on the unit sphere $S^{n-1}$. The measure on $\\mathbb{R}^n$ is named the Euclidean Gaussian moment measure of $\\varphi$. The central result is an existence theorem: for a Borel measure on $\\mathbb{R}^n$ satisfying mild and natural conditions, there is a convex function whose Euclidean Gaussian moment measure is exactly the given measure. This matters because it extends the classical question \"can a shape be recovered from its boundary data?\" to unbounded graphs of convex functions in Gaussian-weighted space.","feed_headline":"Every nice measure is a Gaussian moment of some convex function","feed_subtitle":"A first-variation formula converts prescribed Gaussian data into the existence of a convex epigraph.","key_machinery":"The load-bearing mechanism is the perturbation of a convex function by infimal convolution, $(\\varphi \\Box h)(x)=\\inf_y(\\varphi(y)+h(x-y))$, used inside the Gaussian volume functional of the epigraph. Under such perturbations, the paper proves a variational formula for the Gaussian volume, and the derivative term is recognized as a Borel measure on $\\mathbb{R}^n$—the Euclidean Gaussian moment measure—together with a companion Borel measure on $S^{n-1}$. That formula converts the geometric recovery problem into an analytic existence problem: given the desired measure, find a critical point of a suitable functional, and show that the critical point is a convex function whose moment measure is the prescribed one.","core_discovery":"On the paper's own terms, the central discovery is that the Gaussian Minkowski problem for epigraphs is solvable: the natural necessary conditions on the prescribed measure are also sufficient. Starting from a variational formula that expresses the first variation of the Gaussian volume of the epigraph under an infimal-convolution perturbation as a Borel measure, the paper defines the Euclidean Gaussian moment measure $\\mu_\\varphi$ of a convex function $\\varphi$. It then shows that every Borel measure on $\\mathbb{R}^n$ meeting the stated mild and natural conditions arises as $\\mu_\\varphi$ for some convex $\\varphi$. Thus the Gaussian-weighted boundary data of an epigraph determine the convex function itself, in the same spirit as the classical Minkowski problem determines a convex body from its surface-area measure.","pith_inferences":["Because the proof is variational, the same infimal-convolution perturbation may apply to other weighted volume functionals, such as $L^p$-Gaussian versions, although the paper does not treat those cases.","The spherical measure on $S^{n-1}$ suggests defining a spherical Gaussian Minkowski problem for convex functions, parallel to the Euclidean problem solved here.","The existence theorem alone does not address uniqueness; a plausible extension would ask whether the recovering convex function is unique up to adding affine functions when the prescribed measure has positive density.","If the variational formula holds for a broader class of non-smooth convex functions, the same framework could yield solutions under weaker conditions than those stated in the paper."],"forward_implications":["Given any Borel measure in the stated class, a convex function exists whose Euclidean Gaussian moment measure is the prescribed measure, so the sufficiency part of this Gaussian Minkowski problem is settled.","The variational route supplies a constructive path to the solution, rather than only an abstract compactness argument.","The companion measure on $S^{n-1}$ opens a spherical counterpart of the Euclidean Gaussian moment measure within the same variational framework.","The result transplants the classical shape-recovery question to unbounded convex sets, where the role of the boundary is played by the graph of $\\varphi$.","Further questions of uniqueness, regularity, and stability of the recovering convex function become natural next steps now that existence is established."],"supporting_citations":[],"fun_headline_variants":["Gaussian Minkowski problem solved for convex epigraphs","Convex epigraphs realize any Gaussian moment measure","Prescribed Gaussian data determine a convex epigraph","Nice measures are Gaussian moments of convex functions","First-variation formula yields convex epigraph from Gaussian measure"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that when a convex function is perturbed slightly, the change in the Gaussian volume of its epigraph is exactly captured by a Borel measure; if that derivative fails to exist or to be represented by such a measure for the allowed functions, the existence proof collapses.","fun_headline_variants_meta":{"raw":{"variants":["Gaussian Minkowski problem solved for convex epigraphs","Convex epigraphs realize any Gaussian moment measure","Prescribed Gaussian data determine a convex epigraph","Nice measures are Gaussian moments of convex functions","First-variation formula yields convex epigraph from Gaussian measure"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00068,"raw_usage":{"total_tokens":3015,"prompt_tokens":798,"completion_tokens":2217,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":414,"completion_tokens_details":{"reasoning_tokens":2140}},"tokens_in":414,"tokens_out":2217,"duration_ms":15761,"temperature":1.0,"reasoning_tokens":2140,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:24:55.096665+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a specific finite Borel measure on $\\mathbb{R}^n$ that satisfies the paper's stated conditions but is concentrated on a set where no convex epigraph's Gaussian moment measure could concentrate, and check whether the paper's variational construction still yields a solution; a counterexample would refute the sufficiency claim. A more direct check is to compute the Gaussian volume derivative for an explicit one-dimensional convex function, such as $\\varphi(x)=x^2$, under the infimal-convolution perturbation, and verify that the derivative equals the proposed Euclidean Gaussian moment measure evaluated on the perturbation direction.","supporting_citations":[],"review_version":2}