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REVIEW 3 major objections 4 minor 46 references

The Gaussian Minkowski-type problems for $C$-pseudo-cones

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict Genuine extension of Gaussian Minkowski theory to unbounded C-pseudo-cones with sound core results; abstract overstates uniqueness and a few details need tightening. read the letter →

arxiv 2501.16449 v3 pith:WAT3P4R5 submitted 2025-01-27 math.MG math.PR

Junjie Shan , Wenchuan Hu , Wenxue Xu This is my paper · ORCID
classification math.MGmath.PR MSC 52A4052A38
keywords GaussianMinkowskiproblemC-pseudo-conessurfaceareameasureconelog-MinkowskiWulffshapesvariationalmethodsunboundedconvexsets
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 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.

What carries the argument

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.

What would settle it

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 $\bar h_K + t f$.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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 $1/n$.

Reading between the lines

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

  • 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.
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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 / 4 minor

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}.

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 (3)
  1. [Section 3, Lemma 3.5] 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.
  2. [Section 4, proof of Theorem 4.1] 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.
  3. [Abstract and Section 1] 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.
minor comments (4)
  1. [Section 3, Lemma 3.7 proof] 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).
  2. [Section 7, proof of Theorem 7.3] The reference to 'Lemma 4.3' for the lower bound L_{μ_i}(h̄_{K_i}) > a should be Lemma 7.2.
  3. [Section 5, Theorem 5.6] 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.
  4. [Section 1, after Theorem 1.3] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the existence theorems are variational constructions whose Euler–Lagrange equations are derived from the definition of the Gaussian measures, not assumed; the load-bearing radial variational formulas are external lemmas from Schneider.

full rationale

The derivation chain is self-contained modulo external tools. The Gaussian surface area and cone measures are introduced by explicit integral formulas (3.1) and (6.1). The existence proofs in Lemmas 4.2 and 7.1 maximize the functionals I_mu(f)=gamma_n([f])∫ f dμ and L_mu(f)=gamma_n([f])exp∫ log f dμ; the target measure μ enters only as the data defining the functional, and the final equations μ = c S_gamma_n(K, ·) and μ = C_gamma_n(K, ·)/gamma_n(K) are first-order conditions derived from Lemmas 3.7 and 6.8, respectively, not assumptions imposed on the solution. The derivative computations rely on Lemma 3.5 and Lemma 6.6, quoted from Schneider [37]; this is an external citation dependency and a possible correctness risk if the variational formulas were not valid for the unbounded class K(C,ω), but it is not circular. The only self-citations are [39], used in Lemma 3.8 for the standard continuity of Gaussian volume of convex bodies after truncation, and [42], used in Lemma 5.3 for a support-function identity; both are standard ancillary facts with independent content and do not define or presuppose the target results. The uniqueness arguments use the external Ehrhard inequality, and the non-uniqueness examples are explicit two-point constructions. No fitted parameter is renamed as a prediction, and no theorem is equivalent to its input by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper defines two new measures but introduces no new primitive entities. Its central claims rest on external theorems (Schneider's variational formula and selection theorem, Ehrhard inequality) and the standard Wulff shape construction; these are treated as axioms. No free parameters are fitted.

assumptions (4)
  • domain assumption Radial variational formula for Wulff shapes over K(C,omega): d rho_[h_t](v)/dt|_{t=0} = f(alpha_K(v)) rho_K(v) / hbar_K(alpha_K(v))
    Quoted from Schneider [37] as Lemma 3.5. Used in Lemmas 3.6, 3.7, 6.7 and hence in all existence proofs; assumed without proof.
  • domain assumption Selection theorem for C-pseudo-cones: a sequence with distances to origin bounded away from 0 and infinity has a convergent subsequence
    Quoted from Schneider [35] as Lemma 2.2. Used for compactness in Lemmas 4.2 and 7.1 and the approximation arguments.
  • domain assumption Ehrhard inequality with equality case: for C-pseudo-cones K,L, Phi^{-1}(gamma_n((1-t)K+tL)) >= (1-t)Phi^{-1}(gamma_n(K)) + tPhi^{-1}(gamma_n(L)), with equality iff K = L
    Quoted from Shenfeld-van Handel [40] as Theorem 5.1. Used to derive the log-concavity inequality and the Minkowski inequality behind uniqueness.
  • domain assumption Continuity of Gaussian volume of convex bodies under Hausdorff convergence
    Cited to [39] in Lemma 3.8; a standard fact. Note [39] is by the present authors.

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Pith. "Pith review of The Gaussian Minkowski-type problems for $C$-pseudo-cones." pith.science (2026). https://pith.science/paper/WAT3P4R5

@misc{pith2026250116449,
  author       = {Pith},
  title        = {Pith review of: The Gaussian Minkowski-type problems for $C$-pseudo-cones},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WAT3P4R5}},
  note         = {Machine review of arXiv:2501.16449}
}
abstract

The Gaussian surface area measure and the Gaussian cone measure for $C$-pseudo-cones are introduced and their corresponding Gaussian Minkowski problem and Gaussian log-Minkowski problem are proposed, respectively. The existence and uniqueness of solutions to these problems for $C$-pseudo-cones are established.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

46 extracted references · 39 canonical work pages

  1. [37]

    Schneider, Weighted cone-volume measures of pseudo-cones , Acta Math

    R. Schneider, Weighted cone-volume measures of pseudo-cones , Acta Math. Sci. Ser. B (Engl. Ed.) 45 (2025), no.1, 40–51

    work page 2025

  2. [1]

    W. Ai, Y. Yang, D. Ye, The Lp dual Minkowski problem for unbounded closed convex sets, (2024), arXiv:2404.09804v1

    arXiv 2024

  3. [2]

    B¨ or¨ oczky, P

    K.J. B¨ or¨ oczky, P. Heged˝ us, G. Zhu,On the discrete logarithmic Minkowski problem, Int. Math. Res. Not. (2016), no. 6, 1807–1838

    work page 2016

  4. [3]

    B¨ or¨ oczky, M

    K.J. B¨ or¨ oczky, M. Henk,Cone-volume measures of general centered convex bod- ies, Adv. Math. 286 (2016), 703–721. 28 JUNJIE SHAN, WENCHUAN HU, WENXUE XU ∗

    work page 2016

  5. [4]

    B¨ or¨ oczky, M

    K.J. B¨ or¨ oczky, M. Henk,Cone-volume measure and stability , Adv. Math. 306 (2017), 24–50

    work page 2017

  6. [5]

    B¨ or¨ oczky, E

    K.J. B¨ or¨ oczky, E. Lutwak, D. Yang, G. Zhang,The logarithmic Minkowski prob- lem, J. Amer. Math. Soc. 26 (2013), 831–852

    2013

  7. [6]

    S. Chen, S. Hu, W. Liu, Y. Zhao, On the planar Gaussian-Minkowski problem , Adv. Math. 435 (2023), Paper No. 109351, 32pp

    work page 2023

  8. [7]

    S. Chen, Q. Li, G. Zhu, The logarithmic Minkowski problem for nonsymmetric measures, Trans. Amer. Math. Soc. 371 (2019), no. 4, 2623–2641

    work page 2019

Show all 46 references
  1. [8]

    Cianchi, N

    A. Cianchi, N. Fusco, F. Maggi, A. Pratelli, On the isoperimetric deficit in Gauss space, Amer. J. Math. 133 (2011), 131–186

    2011

  2. [9]

    Ehrhard, Sym´ etrisation dans l’espace de Gauss, Math

    A. Ehrhard, Sym´ etrisation dans l’espace de Gauss, Math. Scand. 53 (1983), 281– 301

    1983

  3. [10]

    Eskenazis, G

    A. Eskenazis, G. Moschidis, The dimensional Brunn-Minkowski inequality in Gauss space, J. Funct. Anal. 280 (2021), Paper No. 108914, 19pp

    2021

  4. [11]

    Y. Feng, S. Hu, L. Xu, On the Lp Gaussian Minkowski problem , J. Differ. Equa. 363 (2023), 350–390

    2023

  5. [12]

    Y. Feng, W. Liu, L. Xu, Existence of non-symmetric solutions to the Gaussian Minkowski problem, J. Geom. Anal. 33 (2023), Paper No. 89, 39pp

    2023

  6. [13]

    Gardner, Geometric Tomography, Cambridge Univ

    R.J. Gardner, Geometric Tomography, Cambridge Univ. Press, Cambridge, 2nd edition, 2006

    2006

  7. [14]

    Gardner, A

    R.J. Gardner, A. Zvavitch, Gaussian Brunn-Minkowski inequalities , Trans. Amer. Math. Soc. 362 (2010), 5333–5353

    2010

  8. [15]

    Haberl, E

    C. Haberl, E. Lutwak, D. Yang, G. Zhang, The even Orlicz Minkowski problem , Adv. Math. 224 (2010) 2485–2510

    2010

  9. [16]

    Hu, The Gaussian log-Minkowski problem , (2024), arXiv:2401.08427v4

    J. Hu, The Gaussian log-Minkowski problem , (2024), arXiv:2401.08427v4

    2024 arXiv

  10. [17]

    Huang, E

    Y. Huang, E. Lutwak, D. Yang, G. Zhang, Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski pro blems, Acta Math. 216 (2016), 325–388

    2016

  11. [18]

    Huang, D

    Y. Huang, D. Xi, Y. Zhao, The Minkowski problem in Gaussian probability space , Adv. Math. 385 (2021), Paper No. 107769, 36pp

    2021

  12. [19]

    Huang, D

    Y. Huang, D. Yang, G. Zhang, Minkowski problems for Geometric measures , Bull. Amer. Math. Soc. (2025)

    2025

  13. [20]

    Klartag, V

    M. Klartag, V. Milman, Geometry of log-concave functions and measures , Geom. Dedicata 112 (2005), 169–182

    2005

  14. [21]

    Kryvonos, D

    L. Kryvonos, D. Langharst, Weighted Minkowski’s existence theorem and projec- tion bodies, Trans. Amer. Math. Soc. 376 (2023), 8447–8493

    2023

  15. [22]

    Lata/suppress la,On some inequalities for Gaussian measures , In Proceedings of the International Congress of Mathematicians, Vol

    R. Lata/suppress la,On some inequalities for Gaussian measures , In Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pages 813–822. Higher Ed. Press, Beijing, 2002

    2002

  16. [23]

    Lehec, A direct proof of the functional Santal´ o inequality , C

    J. Lehec, A direct proof of the functional Santal´ o inequality , C. R. Math. 347 (2009), 55–58. THE GAUSSIAN MINKOWSKI TYPE PROBLEM FOR C-PSEUDO-CONES 29

    2009

  17. [24]

    N. Li, D. Ye, B. Zhu, The dual Minkowski problem for unbounded closed convex sets, Math. Ann. 388 (2024), 2001–2039

    2024

  18. [25]

    Liu, The Lp Gaussian Minkowski problem , Calc

    J. Liu, The Lp Gaussian Minkowski problem , Calc. Var. Partial Differ. Equ. 61 (2022), 1–23

    2022

  19. [26]

    Livshyts, An extension of Minkowski’s theorem and its applications qu es- tions about projections for measures , Adv

    G.V. Livshyts, An extension of Minkowski’s theorem and its applications qu es- tions about projections for measures , Adv. Math. 356 (2019), Paper No.106803, 40pp

    2019

  20. [27]

    Lutwak, The Brunn-Minkowski-Firey theory I: Mixed volumes and the Minkowski Problem, J

    E. Lutwak, The Brunn-Minkowski-Firey theory I: Mixed volumes and the Minkowski Problem, J. Differential Geom. 38 (1993), 131–150

    1993

  21. [28]

    Lutwak, The Brun-Minkowski-Firey theory II: Affine and geominimal su rface areas, Adv

    E. Lutwak, The Brun-Minkowski-Firey theory II: Affine and geominimal su rface areas, Adv. Math. 118 (1996), 244–294

    1996

  22. [29]

    Lutwak, D

    E. Lutwak, D. Xi, D. Yang, G. Zhang, Chord measures in integral geometry and their Minkowski problem , Comm. Pure Appl. Math. 77 (2024), 3277–3330

    2024

  23. [30]

    Milman, L

    E. Milman, L. Rotem, Complemented Brunn-Minkowski inequalities and isoperimetry for homogeneous and non-homogeneous measure s, Adv. Math. 262 (2014), 867–908. Corrigendum: Adv. Math. 307 (2017), 1378–1379

    2014

  24. [31]

    Nayar, T

    P. Nayar, T. Tkocz, A note on a Brunn-Minkowski inequality for the Gaussian measure, Proc. Amer. Math. Soc. 141 (2013), 4027–4030

    2013

  25. [32]

    Schneider, Convex Bodies: The Brunn-Minkowski Theory , Cambridge Univ

    R. Schneider, Convex Bodies: The Brunn-Minkowski Theory , Cambridge Univ. Press, Cambridge, 2014

    2014

  26. [33]

    Schneider, A Brunn-Minkowski theory for coconvex sets of finite volume , Adv

    R. Schneider, A Brunn-Minkowski theory for coconvex sets of finite volume , Adv. Math. 332 (2018), 199–234

    2018

  27. [34]

    Schneider, Minkowski type theorems for convex sets in cones , Acta Math

    R. Schneider, Minkowski type theorems for convex sets in cones , Acta Math. Hungar. 164 (2021), 282–295

    2021

  28. [35]

    Schneider, Pseudo-cones, Adv

    R. Schneider, Pseudo-cones, Adv. in Appl. Math. 155 (2024), Paper No. 102657

    2024

  29. [36]

    Schneider, A weighted Minkowski theorem for pseudo-cones , Adv

    R. Schneider, A weighted Minkowski theorem for pseudo-cones , Adv. Math. 450 (2024), Paper No. 109760, 26pp

    2024

  30. [38]

    Semenov, Y

    V. Semenov, Y. Zhao, The growth rate of surface area measure for noncompact convex sets with prescribed asymptotic cone , (2024), arXiv:2409.18699v1

    2024 arXiv

  31. [39]

    J. Shan, W. Xu, L. Yin, Lp Blaschke-Santal´ o and Petty projection inequalities in Gaussian space, Arch. Math. 122 (2024), 331–342

    2024

  32. [40]

    Shenfeld, R

    Y. Shenfeld, R. van Handel, The equality cases of the Ehrhard-Borell inequality , Adv. Math. 331 (2018), 339–386

    2018

  33. [41]

    Stancu, The discrete planar L0-Minkowski problem, Adv

    A. Stancu, The discrete planar L0-Minkowski problem, Adv. Math. 167 (2002), 160–174

    2002

  34. [42]

    X. Wang, W. Xu, J. Zhou, B. Zhu, Asymptotic theory of C-pseudo-cones , (2024), arXiv:2410.14962v2

    2024 arXiv

  35. [43]

    Y. Xu, J. Li, G. Leng, Dualities and endomorphisms of pseudo-cones , Adv. in Appl. Math. 142 (2023), Paper No. 102434, 31 pp. 30 JUNJIE SHAN, WENCHUAN HU, WENXUE XU ∗

    2023

  36. [44]

    J. Yang, D. Ye, B. Zhu,On the Lp Brunn-Minkowski theory and the Lp Minkowski problem for C-coconvex sets , Int. Math. Res. Not. (2023), no. 7, 6252–6290

    2023

  37. [45]

    Zhu, The logarithmic Minkowski problem for polytopes , Adv

    G. Zhu, The logarithmic Minkowski problem for polytopes , Adv. Math. 262 (2014), 909–931

    2014

  38. [46]

    Zhang, The Minkowski problem for the non-compact convex set with an as- ymptotic boundary condition , (2024), arXiv:2402.12802v1

    N. Zhang, The Minkowski problem for the non-compact convex set with an as- ymptotic boundary condition , (2024), arXiv:2402.12802v1. School of Mathematics, Sichuan University, Chengdu, Sichu an, 610000, PR China Email address : shanjjmath@163.com, wenchuan@scu.edu.cn School of...

    2024 arXiv

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