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The Gaussian Minkowski problem for epigraphs of convex functions

T0 review · 3 major / 2 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2508.12028 v1 pith:YGNPDWHW submitted 2025-08-16 math.FA math.APmath.MG

Xiao Li , Deping Ye This is my paper · ORCID
classification math.FAmath.APmath.MG MSC 52A4152A38
keywords GaussianMinkowskiproblemconvexfunctionsepigraphvolumeinfimalconvolutionmomentmeasurevariationalformulaBorel
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 2 minor

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.

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 (3)
  1. [Abstract (variational formula)] 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.
  2. [Full text (all displayed mathematics)] 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.
  3. [Abstract (problem statement)] 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.
minor comments (2)
  1. [Abstract (terminology)] 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.
  2. [General] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity detected: the abstract reports a genuine variational derivation and an existence theorem with no fitted inputs or self-referential definitions.

full rationale

The abstract describes a variational formula obtained by combining the Gaussian volume of an epigraph with a perturbation via infimal convolution. From this formula a Borel measure on R^n is defined and named the Euclidean Gaussian moment measure. The subsequent Minkowski-type problem asks for a convex function whose Gaussian moment measure equals a pre-given Borel measure, and the abstract claims this problem is solved under mild natural conditions. This is a standard forward direction: a geometric measure is defined from a function, and an inverse problem is posed and solved. There is no indication that the pre-given measure is fitted from the same formula, nor that the existence theorem is assumed by construction. The variational formula itself is presented as a derivation, not as a restatement of the target theorem. No self-citations are visible in the abstract, and no load-bearing argument is reduced to an unverified external citation. The supplied full text is heavily corrupted, preventing equation-level inspection, but the absence of any quoted circular reduction means the honest finding is no circularity. The reader's concern about the unverified first variation is a correctness risk, not a circularity, because circularity requires showing that the derivation uses its own conclusion as an input. That has not been exhibited. Score 0.

Assumptions & free parameters 0 free parameters · 2 assumptions · 1 invented entities

The central claim rests on the standard theory of Gaussian measures and convex analysis, plus a domain assumption about the variational formula. No free parameters are fitted to data. The Euclidean Gaussian moment measure is a newly introduced object without external falsifiable handles.

assumptions (2)
  • standard math The Gaussian measure on R^{n+1} is used to weight the epigraph volume; its properties are standard.
    The epigraph of a convex function is a subset of R^{n+1}; Gaussian volume is the measure with density (2π)^(-(n+1)/2) exp(-|x|^2/2) dx. No parameters are fitted.
  • domain assumption Infimal convolution preserves convexity and is an admissible perturbation for the variational formula.
    The abstract states the perturbation is via infimal convolution; the proof assumes this perturbation gives a first variation of the Gaussian volume. The precise conditions are not given in the abstract.
invented entities (1)
  • Euclidean Gaussian moment measure
    purpose: A Borel measure on R^n associated to a convex function φ, used to formulate and solve the new Minkowski problem.
    New mathematical object defined in the paper. No external falsifiable handle beyond the theorems proved about it; it is introduced as the central object of study.

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Cite this review

Pith. "Pith review of The Gaussian Minkowski problem for epigraphs of convex functions." pith.science (2026). https://pith.science/paper/YGNPDWHW

@misc{pith2026250812028,
  author       = {Pith},
  title        = {Pith review of: The Gaussian Minkowski problem for epigraphs of convex functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YGNPDWHW}},
  note         = {Machine review of arXiv:2508.12028}
}
abstract

A variational formula is derived by combining the Gaussian volume of the epigraph of a convex function $\varphi$ and the perturbation of $\varphi$ via the infimal convolution. This formula naturally leads to a Borel measure on $\mathbb{R}^n$ and a Borel measure on the unit sphere $S^{n-1}$. The resulting Borel measure on $\mathbb{R}^n$ will be called the Euclidean Gaussian moment measure of the convex function $\varphi$, and the related Minkowski-type problem will be studied. In particular, the newly posed Minkowski problem is solved under some mild and natural conditions on the pre-given measure.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Certified Compilation based on G\"odel Numbers

    cs.PL 2025-08 reject novelty 4.0 of 10

    No supported discovery is present: the abstract claims Goedel-number certificates for verified compilation, but the submitted body is a different paper on convex geometry by other authors.

Reference graph

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