{"id":"f8648510-b705-4162-805d-3c3278a1edc4","arxiv_id":"2501.16449","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"New existence and conditional uniqueness theorems for Gaussian Minkowski and log-Minkowski problems on C-pseudo-cones.","lead":"This paper introduces Gaussian-weighted surface area and cone measures for C-pseudo-cones, unbounded convex sets asymptotic to a cone. It proves every finite measure on the relevant direction set is realized up to normalization, with uniqueness when Gaussian volumes match, and shows the realization is generally non-unique.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Existence theorems hinge on the cited radial variational formula (Lemma 3.5); its transfer to unbounded C-pseudo-cones is unproved in the manuscript and should be verified independently.","rationale":"The reader's weakest-assumption analysis points to Lemma 3.5, and my review agrees that this cited radial variational formula is the single external premise on which the existence results depend. I did not find an internal inconsistency in the variational maximizer arguments once that lemma is granted; the compactness arguments, weak continuity, and approximation steps are terse but plausible. The uniqueness section contains a separate wording problem already flagged by the reader: the abstract claims uniqueness without the equal-Gaussian-volume condition of Theorem 5.5, and Theorem 1.5 explicitly gives non-uniqueness. That overstatement does not affect the existence theorems. The recommended verdict remains conditional: the paper should either prove or precisely cite the full Lemma 3.5 for K(C,ω), and the uniqueness claims in the abstract should be qualified.","tokens_in":21869,"tokens_out":55237,"duration_ms":531768,"concrete_test":"Independently re-derive Lemma 3.5 for K(C,ω) from the explicit Wulff-shape radial formula ρ_[h_t](v)=sup_{u∈ω} h_t(u)/(-⟨v,u⟩), checking the a.e. derivative and the uniform bound |ρ_[h_t](v)-ρ_K(v)|≤M|t| for all v∈Ω_C and |t|≤δ. Then rerun the dominated-convergence step of Lemma 3.6 with that bound; if the derivative formula or the uniform M fails for some compact ω⊂Ω, the central existence theorems collapse.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The existence arguments for both the Gaussian Minkowski problem and the Gaussian log-Minkowski problem are driven by Lemma 3.5, quoted from Schneider [37]. Lemma 3.6, Lemma 3.7, Lemma 6.7, and Lemma 7.1 each differentiate a Gaussian volume functional through the radial derivative dρ_[h_t](v)/dt|_{t=0}=ρ_K(v)f(α_K(v))/hbar_K(α_K(v)) and through the Lipschitz bound |ρ_[h_t](v)-ρ_K(v)|≤M|t|. If either the pointwise formula or the uniform dominated-convergence constant fails on the unbounded class K(C,ω), the variational maximizer arguments in Lemma 4.2 and Lemma 7.1 do not yield the claimed solutions, and Theorems 1.2 and 1.4 are unsupported. This is a citation dependency rather than a demonstrated internal inconsistency, but it is the most load-bearing premise in the paper: all existence proofs reduce to it. The rest of the variational and approximation structure appears coherent, and the non-uniqueness constructions are self-contained.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the Gaussian surface area measure S_{γ_n}(K, ·) and the Gaussian cone measure C_{γ_n}(K, ·) for C-pseudo-cones, and studies the associated Gaussian Minkowski and Gaussian log-Minkowski problems. The main results are: Theorem 1.2, every nonzero finite Borel measure on Ω_{C^o} is, up to a normalizing factor c, the Gaussian surface area measure of some C-pseudo-cone; Theorem 1.3, uniqueness holds when the two cones have equal Gaussian volume; Theorem 1.4, every nonzero finite Borel measure is the normalized Gaussian cone measure of some C-pseudo-cone; and Theorem 1.5, both problems have non-unique solutions in general. The proofs combine variational formulas for Gaussian volumes of Wulff shapes in cones, Schneider's results on pseudo-cones, Ehrhard's inequality, and approximation arguments over exhausted compact subsets of Ω_{C^o}.","tokens_in":22093,"tokens_out":19110,"duration_ms":186799,"significance":"If the results hold, the paper gives a substantial extension of the Gaussian Minkowski theory from convex bodies to unbounded convex sets with prescribed recession cone. The introduction of the Gaussian cone measure, the observation that the log-Minkowski problem on C-pseudo-cones requires no subspace concentration condition, and the explicit constructions of non-uniqueness and of measures that are not surface area measures are valuable contributions. The proofs are largely self-contained and are organized around clear variational functionals; the non-uniqueness examples in Sections 5 and 7 are concrete and convincing. The main caveat is that the entire existence machinery rests on a variational formula quoted from Schneider's work, so the manuscript should make that dependency fully checkable.","major_comments":[{"comment":"The variational engine of the paper is quoted from [37] without the precise statement being given. Lemma 3.5(a) is used in Lemma 3.6 and Lemma 6.7, and the uniform Lipschitz bound in Lemma 3.5(b) is essential for the dominated-convergence steps in those proofs and hence for Theorems 4.1 and 7.3. Please state the exact theorem or lemma number in [37], confirm that the class K(C, ω) satisfies all hypotheses of that result, and explain how the uniform bound in part (b) is obtained. This is not a request for a new proof, but without this information the central existence claims cannot be independently checked.","section":"Section 3, Lemma 3.5"},{"comment":"The step asserting that the normalization constants c_i are uniformly bounded is too terse. The lower bound γ_n(K_i) ≥ γ_n(z + C) requires the existence of a common point z lying in all K_i; this should be derived explicitly from the uniform lower bound 0 < m < dist(o, ∂K_i) and the radial structure of C-pseudo-cones. The numerator bound also needs a uniform estimate such as h̄_{K_i}(u) ≤ −⟨z, u⟩ for u ∈ Ω. As written, this is the least documented step in the approximation proof and should be expanded.","section":"Section 4, proof of Theorem 4.1"},{"comment":"The abstract and the introduction state that existence and uniqueness of solutions are established. This is unconditionally false in view of Theorem 1.5 (and Theorems 5.6 and 7.4), which construct distinct solutions to both problems. Please rephrase the claim to indicate that uniqueness holds only under additional assumptions, namely equal Gaussian volume in Theorem 1.3 or the volume constraints discussed in Section 5.","section":"Abstract and Section 1"}],"minor_comments":[{"comment":"The line γ_n(C) < 1/2 is false when n = 1, since a pointed cone in R is a closed half-line with Gaussian measure exactly 1/2. Either assume n ≥ 2 or replace the strict inequality by ≤; the argument only uses the identity γ_n(L) + V_G(L) = γ_n(C).","section":"Section 3, Lemma 3.7 proof"},{"comment":"The reference to 'Lemma 4.3' for the lower bound L_{μ_i}(h̄_{K_i}) > a should be Lemma 7.2.","section":"Section 7, proof of Theorem 7.3"},{"comment":"The notation in the proof is introduced abruptly: it should be stated explicitly that one fixes b ∈ ω and then applies a rotation sending b to −e_n, and that rotational invariance of the Gaussian measure justifies this reduction.","section":"Section 5, Theorem 5.6"},{"comment":"The statement that the volume restriction in Theorem 1.3 can be relaxed to γ_n(K), γ_n(L) ≤ (1/2)γ_n(C) in discrete cases is not proved in general. It is supported only by the planar example in Remark 5.8; please label it as an observation for that example or provide a proof in the stated generality.","section":"Section 1, after Theorem 1.3"}],"recommendation":"major_revision","confidential_remarks":"The paper is suitable in scope for a convex geometry journal. The dependence on Schneider's variational formula [37] is legitimate, but because it is the load-bearing external input, I recommend that the editor have a referee familiar with [37] confirm that Lemma 3.5 is stated exactly as used. The main revision tasks are to make the uniform bound on c_i fully explicit and to correct the overstatement of uniqueness in the abstract and introduction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuine extension of the Gaussian Minkowski program to unbounded C-pseudo-cones. It is not just Schneider's homogeneous-weight construction with a new weight: the Gaussian density is non-homogeneous, so the normalization constant cannot be absorbed, and the authors introduce a new variational functional (Gaussian volume times ∫ h dμ) because the standard covolume trick does not work. The main existence theorems (1.2 and 1.4) are proved in detail, and the non-uniqueness constructions in Theorems 1.5/5.6/7.4 are self-contained and convincing.\n\nThe paper is honest about the scope: uniqueness is only under equal Gaussian volume (Theorem 1.3), and both problems admit distinct solutions (Theorem 1.5). The abstract, however, says 'uniqueness ... are established' without the caveat. That is too strong and should be revised to 'conditional uniqueness' or similar.\n\nThe soft spots are mostly minor. Lemma 3.7 asserts γ_n(C) < 1/2 for pointed cones; for n=1 a ray has Gaussian measure exactly 1/2. Presumably the paper intends n≥2, and the statement is not needed for the main results. In Theorem 4.1, the uniform boundedness of the normalization constants c_i is asserted tersely; it is probably correct because the K_i contain a fixed translate of C and their support functions are controlled, but the proof should give one sentence. The largest dependency is Lemma 3.5, Schneider's radial variational formula for Wulff shapes. Every existence proof differentiates through it. The authors cite it rather than prove it, which is normal, but a referee should verify that the hypotheses of that lemma indeed cover the class K(C,ω) with the Gaussian volume. I have no concrete reason to think it fails—Schneider proved it for exactly this setting—but it is the load-bearing wall.\n\nOverall the central claims look correct and the paper advances the theory of Minkowski problems for unbounded convex sets. Convex geometers working on Gaussian or weighted Minkowski problems will want it. It deserves a serious referee; I would send it out with minor revision requests: fix the abstract, add n≥2, and expand the c_i bound.","headline":"Genuine extension of Gaussian Minkowski theory to unbounded C-pseudo-cones with sound core results; abstract overstates uniqueness and a few details need tightening.","tokens_in":22598,"tokens_out":5354,"would_cite":true,"duration_ms":49331,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A40","52A38"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every nonzero finite Borel measure on the dual sphere of a pointed cone is, up to normalization, the Gaussian surface area measure of some C-pseudo-cone, and the normalized Gaussian cone measure represents the same measures.","keywords":["Gaussian Minkowski problem","C-pseudo-cones","Gaussian surface area measure","Gaussian cone measure","log-Minkowski problem","Wulff shapes","variational methods","unbounded convex sets"],"falsifier":"Exhibit a $C$-pseudo-cone $K \\in \\mathcal K(C,\\omega)$ and a continuous $f:\\omega\\to\\mathbb R$ for which the identity $d\\rho_{[h_t]}(v)/dt|_{t=0} = f(\\alpha_K(v))\\rho_K(v)/\\bar h_K(\\alpha_K(v))$ fails on a set of positive measure, or for which the bound $|\\rho_{[h_t]}(v)-\\rho_K(v)| \\le M|t|$ fails; either outcome would break the variational existence proof. A concrete numerical test is to compute both sides of that derivative identity for a pointed cone with a non-smooth boundary and a Wulff shape with support function $\\bar h_K + t f$.","tokens_in":1949,"feed_emoji":"📐","tokens_out":4934,"duration_ms":91191,"temperature":0.7,"pith_summary":"The paper extends the classical Minkowski problem to unbounded convex sets, namely C-pseudo-cones, by weighting their boundary with the Gaussian density. It introduces two measures on the spherical part of the dual cone, the Gaussian surface area measure and the Gaussian cone measure, and formulates the corresponding Gaussian Minkowski and Gaussian log-Minkowski problems. The main results assert that every nonzero finite Borel measure on that sphere is, up to an explicit normalization, such a surface-area measure of some C-pseudo-cone, and that every such measure is the normalized Gaussian cone measure of some C-pseudo-cone. Uniqueness holds when the two cones have equal Gaussian volume, while distinct cones with the same measures exist in general. If correct, this provides a complete Gaussian-weight existence theory for the noncompact analogue of the Minkowski problem, with the log version needing no subspace concentration condition.","feed_headline":"Every finite Borel measure becomes Gaussian surface area","feed_subtitle":"Gaussian Minkowski and log-Minkowski problems on unbounded convex sets: existence, uniqueness under equal volumes, non-uniqueness otherwise.","key_machinery":"The central object is the Wulff shape $[h]$ inside the cone $C$, defined as the intersection of $C$ with halfspaces $\\{y : \\langle y,u\\rangle \\le -h(u)\\}$ for $u$ in a compact set $\\omega \\subset \\Omega_{C^\\circ}$. Two variational functionals are maximized over these shapes: $I_\\mu(f)=\\gamma_n([f])\\int_\\omega f\\,d\\mu$ for the Gaussian Minkowski problem and $L_\\mu(f)=\\gamma_n([f])\\exp\\int_\\omega \\log f\\,d\\mu$ for the log-Minkowski problem. The derivative engine is the radial variational formula of Lemma 3.5, which converts the first-order change of the Wulff shape into $f(\\alpha_K(v))\\rho_K(v)/\\bar h_K(\\alpha_K(v))$, and the Ehrhard inequality with equality case supplies the uniqueness conclusion.","core_discovery":"The central claim is Theorem 1.2: for any nonzero finite Borel measure $\\mu$ on $\\Omega_{C^\\circ}$, there is a $C$-pseudo-cone $K$ with $\\mu = c\\,S_{\\gamma_n}(K,\\cdot)$, where $c = \\int_{\\Omega} \\bar h_K\\,d\\mu / \\gamma_n(K)$. Theorem 1.4 is the log version: the normalized Gaussian cone measure satisfies $C_{\\gamma_n}(K,\\cdot)/\\gamma_n(K) = \\mu$. Theorem 1.3 says that within the $C$-determined class $\\mathcal K(C,\\omega)$, equal Gaussian volume $\\gamma_n(K)=\\gamma_n(L)$ together with equal Gaussian surface area measures forces $K=L$. The authors also construct distinct cones, using one-directional translates of hyperplane sections, that share the same Gaussian surface area measure or the same Gaussian cone measure, showing the volume condition cannot be dropped.","pith_inferences":["The same variational functionals should extend to other non-homogeneous weightings; replacing $e^{-|x|^2/2}$ by $e^{-|x|^p/p}$ would yield $L^p$-Gaussian Minkowski problems for $C$-pseudo-cones, provided the radial variational lemma survives.","The absence of subspace concentration in the log-Minkowski theorem suggests that the unbounded cone $C$, rather than the measure, absorbs the mass that would otherwise concentrate; taking a limit where $C$ approaches a halfspace may recover the convex-body obstruction.","The atom-size thresholds in Remarks 5.7 and 7.5 are natural candidates for the exact representability range; testing whether every finite measure with all atoms below the threshold is representable would settle the sharpness question.","The planar threshold $\\gamma_2(K),\\gamma_2(L) \\le \\gamma_2(C)/2$ for uniqueness suggests a general threshold phenomenon, and computing the analogous constant in higher dimensions would test whether the phenomenon persists."],"forward_implications":["Every nonzero finite Borel measure on $\\Omega_{C^\\circ}$ is, up to the explicit constant $c$, the Gaussian surface area measure of a $C$-pseudo-cone, giving a Gaussian analogue of the weighted Minkowski theorem for unbounded convex sets.","The same measure class is realized by normalized Gaussian cone measures, so the Gaussian log-Minkowski problem for $C$-pseudo-cones has solutions for all finite nonzero measures, with no subspace concentration condition.","If two $C$-determined pseudo-cones have equal Gaussian volume and equal Gaussian surface area measures, then they coincide; without the volume assumption, distinct solutions exist.","Distinct $C$-pseudo-cones can share the same Gaussian cone measure, and measures with a single sufficiently large atom cannot be represented by any $C$-pseudo-cone.","The Gaussian cone measure is strictly less than the Gaussian cone volume of the union of segments from the origin to the boundary, differing at least by the factor $1/n$."],"supporting_citations":[{"why":"Supplies Lemma 3.5, the radial variational formula for Wulff shapes in cones that drives every derivative computation.","marker":"[37]"},{"why":"Establishes the homogeneous weighted Minkowski theorem for pseudo-cones whose finite-measure paradigm the Gaussian measures inherit.","marker":"[36]"},{"why":"Introduces pseudo-cones and the selection theorem used to extract convergent maximizer sequences.","marker":"[35]"},{"why":"Solves the Gaussian Minkowski problem for convex bodies and supplies the comparison used for non-uniqueness and volume thresholds.","marker":"[18]"},{"why":"Provides the Ehrhard inequality with equality case that yields uniqueness under equal Gaussian volumes.","marker":"[40]"},{"why":"Gives the Minkowski-type theorem for convex sets in cones that motivates the finite-measure setting.","marker":"[34]"},{"why":"Supplies the continuity of Gaussian volume for convex bodies invoked in Lemma 3.8.","marker":"[39]"}],"fun_headline_variants":["Any finite Borel measure is a Gaussian surface area","Gaussian Minkowski problems solved for C-pseudo-cones","Every measure becomes Gaussian cone or surface measure","Uniqueness when Gaussian volumes match in cone problems","Log-Minkowski: Gaussian cone measure matches any measure"],"cache_read_input_tokens":24832,"weakest_assumption_plain":"The whole proof leans on the cited formula saying that a small change in the support function changes the radial function of a Wulff shape in a specific linear way; if that formula, or the bound that lets the derivative pass under the Gaussian integral, fails for some C-determined pseudo-cone, the maximizer arguments stop working.","fun_headline_variants_meta":{"raw":{"variants":["Any finite Borel measure is a Gaussian surface area","Gaussian Minkowski problems solved for C-pseudo-cones","Every measure becomes Gaussian cone or surface measure","Uniqueness when Gaussian volumes match in cone problems","Log-Minkowski: Gaussian cone measure matches any measure"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000432,"raw_usage":{"total_tokens":2120,"prompt_tokens":777,"completion_tokens":1343,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":393,"completion_tokens_details":{"reasoning_tokens":1265}},"tokens_in":393,"tokens_out":1343,"duration_ms":13051,"temperature":1.0,"reasoning_tokens":1265,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T13:13:59.765682+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a $C$-pseudo-cone $K \\in \\mathcal K(C,\\omega)$ and a continuous $f:\\omega\\to\\mathbb R$ for which the identity $d\\rho_{[h_t]}(v)/dt|_{t=0} = f(\\alpha_K(v))\\rho_K(v)/\\bar h_K(\\alpha_K(v))$ fails on a set of positive measure, or for which the bound $|\\rho_{[h_t]}(v)-\\rho_K(v)| \\le M|t|$ fails; either outcome would break the variational existence proof. A concrete numerical test is to compute both sides of that derivative identity for a pointed cone with a non-smooth boundary and a Wulff shape with support function $\\bar h_K + t f$.","supporting_citations":[{"cited_title":"Schneider, Weighted cone-volume measures of pseudo-cones , Acta Math","cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 3.5, the radial variational formula for Wulff shapes in cones that drives every derivative computation."},{"cited_title":"Schneider, A weighted Minkowski theorem for pseudo-cones , Adv","cited_arxiv_id":null,"evidence_quote":"Establishes the homogeneous weighted Minkowski theorem for pseudo-cones whose finite-measure paradigm the Gaussian measures inherit."},{"cited_title":"Schneider, Pseudo-cones, Adv","cited_arxiv_id":null,"evidence_quote":"Introduces pseudo-cones and the selection theorem used to extract convergent maximizer sequences."},{"cited_title":"Huang, D","cited_arxiv_id":null,"evidence_quote":"Solves the Gaussian Minkowski problem for convex bodies and supplies the comparison used for non-uniqueness and volume thresholds."},{"cited_title":"Shenfeld, R","cited_arxiv_id":null,"evidence_quote":"Provides the Ehrhard inequality with equality case that yields uniqueness under equal Gaussian volumes."},{"cited_title":"Schneider, Minkowski type theorems for convex sets in cones , Acta Math","cited_arxiv_id":null,"evidence_quote":"Gives the Minkowski-type theorem for convex sets in cones that motivates the finite-measure setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the continuity of Gaussian volume for convex bodies invoked in Lemma 3.8."}],"review_version":1}