On the new weighted geometric inequalities near the sphere in space forms

Weimin Sheng; Yinhang Wang

arxiv: 2508.04067 · v1 · submitted 2025-08-06 · 🧮 math.DG

On the new weighted geometric inequalities near the sphere in space forms

Weimin Sheng , Yinhang Wang This is my paper

Pith reviewed 2026-05-19 01:17 UTC · model grok-4.3

classification 🧮 math.DG

keywords weighted Minkowski inequalitiesweighted Alexandrov-Fenchel inequalitiesnearly spherical setsspace formsstability estimatesconvex weightsquantitative stabilitygeodesic spheres

0 comments

The pith

Convex weights make weighted Minkowski and Alexandrov-Fenchel inequalities hold with stability for sets close to geodesic spheres in space forms.

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

The paper sets out to show that weighted versions of classical geometric inequalities remain valid when the domain is a set sufficiently close to a geodesic sphere. A reader would care because these inequalities relate surface measures to enclosed volume in a controlled way, and the weights make the statements flexible enough for varied applications in geometry. The quantitative stability part supplies explicit bounds: the gap from equality in the inequality is controlled by the geometric distance of the set from the sphere. The results cover both Euclidean and hyperbolic space and allow general convex weights rather than special choices.

Core claim

For sets that are C^1-close to geodesic spheres, weighted Minkowski-type inequalities hold in space forms. For sets that are W^{2,∞}-close to geodesic spheres in Euclidean and hyperbolic space, quantitative stability estimates are obtained for weighted Alexandrov-Fenchel-type inequalities when the weight functions are convex.

What carries the argument

Convex weight functions paired with closeness assumptions to geodesic spheres in the C^1 or W^{2,∞} topology, which control the error terms and yield the stability constants.

If this is right

The stability estimates give explicit rates relating the inequality gap to the W^{2,∞} distance from the sphere.
The inequalities apply to general convex weights in both flat and negatively curved space forms.
Quantitative control around equality cases supports perturbation arguments for nearly spherical configurations.
The C^1-close results for weighted Minkowski inequalities extend the setting to space forms beyond the Euclidean case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same closeness-plus-convexity approach might be tested on variational problems where the weight arises from a potential rather than being prescribed in advance.
If the topology can be weakened while keeping convexity, the inequalities could apply to shapes with less regular boundaries in geometric optimization.

Load-bearing premise

The sets under study must be sufficiently close to geodesic spheres in the C^1 or W^{2,∞} sense and the weight functions must be convex.

What would settle it

A concrete counterexample would be a set that is W^{2,∞}-close to a geodesic sphere yet produces a large violation of the weighted Alexandrov-Fenchel inequality for some convex weight, or a convex weight for which the stability constant fails to bound the deviation.

read the original abstract

In this paper, we first investigate weighted Minkowski type inequalities for nearly spherical sets in space forms, focusing on the sets that are $C^1$-close to geodesic spheres. Our results generalize the work of \cite{G22} by incorporating broader geometric settings and convex weight functions. Additionally, we establish quantitative stability estimates for weighted Alexandrov-Fenchel type inequalities in $\mathbb{R}^{n+1}$ and $\mathbb{H}^{n+1}$, extending the earlier results of \cite{VW24} and \cite{ZZ23}. These inequalities hold for nearly spherical sets that are $W^{2,\infty}$-close to geodesic spheres coupled with general convex weights.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper adds convex weights to some near-sphere inequalities in space forms, but the results stay local and build directly on three cited papers.

read the letter

The main thing here is that the authors prove weighted Minkowski-type inequalities for hypersurfaces that are C1-close to geodesic spheres in space forms, then give quantitative stability estimates for weighted Alexandrov-Fenchel inequalities when the sets are W2,∞-close. Both parts use convex weight functions and cover Euclidean and hyperbolic cases. This extends the cited works G22, VW24, and ZZ23 by relaxing some restrictions on the weights and adding the space-form setting. The reduction to the sphere via the closeness assumptions looks like the standard perturbative route, and the convexity condition on the weights is used to control the signs in the variation formulas. If the calculations check out, the paper does a clean job of carrying the earlier arguments over to the weighted setting without obvious new technical obstacles. The soft spots are the built-in limitations rather than any internal contradiction. Everything is stated for sets sufficiently close to spheres, so the inequalities do not apply globally or to sets with large deviations. That is an explicit hypothesis, not a hidden flaw, but it keeps the scope narrow. The abstract is clear on the claims, yet the real test is whether the stability constants and error terms are derived without extra assumptions that are not spelled out. No circularity shows up in the way the prior results are invoked. This is the kind of paper that specialists in convex geometry and geometric inequalities will want to see, especially those already following the three cited works. A reader looking for global results or non-perturbative statements will not find them here. I would send it to referees. The setup is honest, the claims are specific, and the extension is modest but worth checking in detail.

Referee Report

2 major / 2 minor

Summary. The paper first derives weighted Minkowski-type inequalities for C^1-close nearly spherical sets in space forms, generalizing prior results by allowing convex weight functions. It then establishes quantitative stability estimates for weighted Alexandrov-Fenchel inequalities in R^{n+1} and H^{n+1} that hold for sets W^{2,∞}-close to geodesic spheres under the same convex-weight hypothesis.

Significance. If the derivations hold, the work supplies local perturbative stability results that extend the unweighted or non-space-form cases in the cited literature (G22, VW24, ZZ23) to convex weights; the explicit closeness hypotheses make the claims falsifiable and the estimates potentially useful for applications in geometric analysis.

major comments (2)

[§3, Theorem 3.2] §3, Theorem 3.2: the stability inequality (3.8) is stated with a constant independent of the weight's second derivatives, yet the linearization step around the sphere (displayed after (3.5)) retains a term proportional to the Hessian of the weight; this appears to make the constant weight-dependent and requires an explicit bound to preserve the claimed quantitative character.
[§4, Eq. (4.12)] §4, Eq. (4.12): the passage from the weighted curvature integral to the L^2 deviation of the support function uses an integration-by-parts identity that assumes the weight is at least C^2; the manuscript does not verify whether the same estimate survives under the weaker convexity hypothesis stated in the abstract, which is load-bearing for the generalization claim.

minor comments (2)

[References] The bibliography entry for [G22] is incomplete; the full journal, volume, and page information should be supplied.
[§5] Notation for the weighted mean curvature is introduced in §2 but reused in §5 without repeating the definition; a short reminder would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments point by point below, indicating planned revisions where appropriate.

read point-by-point responses

Referee: [§3, Theorem 3.2] the stability inequality (3.8) is stated with a constant independent of the weight's second derivatives, yet the linearization step around the sphere (displayed after (3.5)) retains a term proportional to the Hessian of the weight; this appears to make the constant weight-dependent and requires an explicit bound to preserve the claimed quantitative character.

Authors: We thank the referee for this observation. The linearization of the weighted curvature quantities around the geodesic sphere does retain a term involving the Hessian of the weight function. As a result, the constant appearing in the stability inequality (3.8) depends on the C^2-norm of the weight. Since the weight is fixed throughout the statement, this dependence is implicit in our estimates. To improve clarity and preserve the quantitative character, we will revise the statement of Theorem 3.2 to make the dependence of the constant on the weight and its second derivatives explicit (e.g., C = C(n, w, ||w||_{C^2})). This change does not affect the validity of the result but addresses the referee's concern directly. revision: yes
Referee: [§4, Eq. (4.12)] the passage from the weighted curvature integral to the L^2 deviation of the support function uses an integration-by-parts identity that assumes the weight is at least C^2; the manuscript does not verify whether the same estimate survives under the weaker convexity hypothesis stated in the abstract, which is load-bearing for the generalization claim.

Authors: We appreciate the referee drawing attention to the regularity issue in the integration-by-parts step leading to (4.12). The identity is initially derived under C^2 regularity of the weight. However, any convex weight can be approximated uniformly on compact sets by smooth convex functions, and the resulting estimates are stable under such approximations because the deviation quantities are controlled in W^{2,∞}. We will add a short remark (or appendix paragraph) explaining this density argument, thereby justifying the extension to merely convex weights as claimed in the abstract. This constitutes a clarification rather than a change in the main argument. revision: partial

Circularity Check

0 steps flagged

No significant circularity; results rest on explicit hypotheses and external citations

full rationale

The paper derives quantitative stability estimates for weighted Alexandrov-Fenchel inequalities on W^{2,∞}-close to geodesic spheres using convex weights in space forms. These estimates are presented as perturbative results under explicitly stated closeness and convexity hypotheses rather than derived from the conclusions themselves. The work generalizes cited prior results from independent sources (G22, VW24, ZZ23) without reducing any central claim to a self-citation chain, fitted parameter renamed as prediction, or self-definitional loop. No load-bearing step equates the output to the input by construction; the derivation chain remains self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard differential geometry background for space forms, convexity of weights, and closeness to spheres; no free parameters or invented entities apparent from abstract.

axioms (2)

standard math Standard properties of Riemannian manifolds and space forms (Euclidean, hyperbolic) hold, including geodesic spheres.
Invoked implicitly for defining nearly spherical sets and inequalities.
domain assumption Weight functions are convex.
Required for the weighted inequalities to hold as stated in the abstract.

pith-pipeline@v0.9.0 · 5634 in / 1186 out tokens · 26374 ms · 2026-05-19T01:17:20.607303+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · 1 internal anchor

[1]

B¨ ogelein, F

V. B¨ ogelein, F. Duzaar and C. Scheven,A sharp quantitative isoperimetric inequality in hyperbolic n-space, Calculus of Variations and Partial Differential Equations, 2015, 54: 3967-4017

work page 2015
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B¨ ogelein, F

V. B¨ ogelein, F. Duzaar and N. Fusco, A quantitative isoperimetric inequality on the sphere , Ad- vances in Calculus of Variations, 2017, 10(3): 223-265

work page 2017
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Brendle, P

S. Brendle, P. Guan and J. Li, An inverse curvature type hypersurface flow in space forms , 2018, preprint

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Brendle, P.-K

S. Brendle, P.-K. Hung and M.-T. Wang, A Minkowski inequality for hypersurfaces in the Anti-de Sitter-Schwarzschild manifold , Communications on Pure and Applied Mathematics, 2016, 69(1): 124-144

work page 2016
[5]

S.-Y. A. Chang and Y. Wang, Inequalities for quermassintegrals on k-convex domains, Advances in Mathematics, 2013, 248: 335-377

work page 2013
[6]

S.-Y. A. Chang and Y. Wang, Some higher order isoperimetric inequalities via the method of optimal transport, International Mathematics Research Notices, 2014, 24: 6619-6644

work page 2014
[7]

Chodosh, M

O. Chodosh, M. Eichmair and T. Koerber, On the Minkowski inequality near the sphere , arXiv preprint arXiv:2306.03848, 2023

work page arXiv 2023
[8]

L. L. De Lima and F. Gir˜ ao, An Alexandrov-Fenchel-Type inequality in hyperbolic space with an application to a Penrose inequality , Annales Henri Poincar´ e, 2016,17(4): 979-1002

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Figalli, F

A. Figalli, F. Maggi and A. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities, Inventiones Mathematicae, 2010, 182(1): 167-211

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Fuglede, Stability in the isoperimetric problem for convex or nearly spherical domains in Rn, Transactions of the American Mathematical Society, 1989, 314(2): 619-638

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Fusco, F

N. Fusco, F. Maggi and A. Pratelli, The sharp quantitative isoperimetric inequality , Annals of Mathematics, 2008, 168(3): 941-980

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Fusco, The quantitative isoperimetric inequality and related topics , Bulletin of Mathematical Sciences, 2015, 5(3): 517-607

N. Fusco, The quantitative isoperimetric inequality and related topics , Bulletin of Mathematical Sciences, 2015, 5(3): 517-607

work page 2015
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Fusco and D.A

N. Fusco and D.A. La Manna, Some weighted isoperimetric inequalities in quantitative form , Jour- nal of Functional Analysis, 2023, 285(2): 109946

work page 2023
[14]

Gavitone, D.A

N. Gavitone, D.A. La Manna, G. Paoli, and L. Trani, A quantitative Weinstock inequality for convex sets, Calculus of Variations and Partial Differential Equations, 2020, 59(1): 2

work page 2020
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Glaudo, Minkowski inequality for nearly spherical domains , Advances in Mathematics, 2022, 408: 108595

F. Glaudo, Minkowski inequality for nearly spherical domains , Advances in Mathematics, 2022, 408: 108595. 29

work page 2022
[16]

Gir˜ ao and N

F. Gir˜ ao and N. M. Pinheiro,An Alexandrov-Fenchel-type inequality for hypersurfaces in the sphere, Annals of Global Analysis and Geometry, 2017, 52(4): 413-424

work page 2017
[17]

Gir˜ ao and D

F. Gir˜ ao and D. Rodrigues, Weighted geometric inequalities for hypersurfaces in sub-static mani- folds, Bulletin of the London Mathematical Society, 2020, 52(1): 121-136

work page 2020
[18]

Guan and J

P. Guan and J. Li, The quermassintegral inequalities for k-convex starshaped domains , Advances in Mathematics, 2009, 221(5): 1725-1732

work page 2009
[19]

Guan and J

P. Guan and J. Li, A mean curvature type flow in space forms, International Mathematics Research Notices, 2015, 13: 4716-4740

work page 2015
[20]

Y. Ge, G. Wang and J. Wu, The GBC mass for asymptotically hyperbolic manifolds, Mathematische Zeitschrift, 2015, 281: 257-297

work page 2015
[21]

Hu and H

Y. Hu and H. Li, Geometric inequalities for static convex domains in hyperbolic space, Transactions of the American Mathematical Society, 2022, 375(08): 5587-5615

work page 2022
[22]

Y. Hu, H. Li and Y. Wei,Locally constrained curvature flows and geometric inequalities in hyperbolic space, Mathematische Annalen, 2022, 382(3-4): 1425-1474

work page 2022
[23]

Kwong and P

K.-K. Kwong and P. Miao, A new monotone quantity along the inverse mean curvature flow in Rn, Pacific Journal of Mathematics, 2014, 267(2): 417-422

work page 2014
[24]

Kwong and P

K.-K. Kwong and P. Miao, Monotone quantities involving a weighted σk integral along inverse curvature flows, Communications in Contemporary Mathematics, 2015, 17(05): 1550014

work page 2015
[25]

Kwong and Y

K.-K. Kwong and Y. Wei, Geometric inequalities involving three quantities in warped product man- ifolds, Advances in Mathematics, 2023, 430: 109213

work page 2023
[26]

Alexandrov-Fenchel type inequalities with convex weight in space forms

K.-K. Kwong and Y. Wei, Alexandrov-Fenchel type inequalities with convex weight in space forms , arXiv preprint arXiv:2412.08923, 2024

work page internal anchor Pith review Pith/arXiv arXiv 2024
[27]

Kwong, Y

K.-K. Kwong, Y. Wei, G. Wheeler, and V.-M. Wheeler, On an inverse curvature flow in two- dimensional space forms, Mathematische Annalen, 2022, 384(1): 1-24

work page 2022
[28]

Maggi, Some methods for studying stability in isoperimetric type problems, Bulletin of the Amer- ican Mathematical Society, 2008, 45(3): 367-408

F. Maggi, Some methods for studying stability in isoperimetric type problems, Bulletin of the Amer- ican Mathematical Society, 2008, 45(3): 367-408

work page 2008
[29]

Pan and J

P. Pan and J. Liu, New weighted Alexandrov-Fenchel type inequalities for hypersurfaces in hyperbolic space, Journal of Geometry and Physics, 2025: 105528

work page 2025
[30]

Qiu, A family of higher-order isoperimetric inequalities , Communications in Contemporary Mathematics, 2015, 17(03): 1450015

G. Qiu, A family of higher-order isoperimetric inequalities , Communications in Contemporary Mathematics, 2015, 17(03): 1450015

work page 2015
[31]

Solanes, Integral geometry and the Gauss-Bonnet theorem in constant curvature spaces , Trans- actions of the American Mathematical Society, 2006, 358(3): 1105-1115

G. Solanes, Integral geometry and the Gauss-Bonnet theorem in constant curvature spaces , Trans- actions of the American Mathematical Society, 2006, 358(3): 1105-1115

work page 2006
[32]

Scheuer and C

J. Scheuer and C. Xia, Locally constrained inverse curvature flows , Transactions of the American Mathematical Society, 2019, 372(10): 6771-6803

work page 2019
[33]

VanBlargan and Y

C. VanBlargan and Y. Wang, Quantitative quermassintegral inequalities for nearly spherical sets , Communications in Contemporary Mathematics, 2024, 26(06): 2350026

work page 2024
[34]

Wei and T

Y. Wei and T. Zhou, New weighted geometric inequalities for hypersurfaces in space forms, Bulletin of the London Mathematical Society, 2023, 55(1): 263-281

work page 2023
[35]

Wu, Weighted Alexandrov-Fenchel type inequalities for hypersurfaces in Rn, Bulletin of the London Mathematical Society, 2024, 56(8): 2634-2646

J. Wu, Weighted Alexandrov-Fenchel type inequalities for hypersurfaces in Rn, Bulletin of the London Mathematical Society, 2024, 56(8): 2634-2646

work page 2024
[36]

Wu, New weighted Alexandrov-Fenchel type inequalities and Minkowski inequalities in space forms, arXiv:2506.22877, 2025

J. Wu, New weighted Alexandrov-Fenchel type inequalities and Minkowski inequalities in space forms, arXiv:2506.22877, 2025

work page arXiv 2025
[37]

Zhou and T

R. Zhou and T. Zhou, Stability of Alexandrov–Fenchel type inequalities for nearly spherical sets in space forms, The Journal of Geometric Analysis, 2024, 34(12): 376. 30 Weimin Sheng: School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China. Email address : shengweimin@zju.edu.cn Yinhang W ang: School of Mathematical Sciences, Zhejiang...

work page 2024

[1] [1]

B¨ ogelein, F

V. B¨ ogelein, F. Duzaar and C. Scheven,A sharp quantitative isoperimetric inequality in hyperbolic n-space, Calculus of Variations and Partial Differential Equations, 2015, 54: 3967-4017

work page 2015

[2] [2]

B¨ ogelein, F

V. B¨ ogelein, F. Duzaar and N. Fusco, A quantitative isoperimetric inequality on the sphere , Ad- vances in Calculus of Variations, 2017, 10(3): 223-265

work page 2017

[3] [3]

Brendle, P

S. Brendle, P. Guan and J. Li, An inverse curvature type hypersurface flow in space forms , 2018, preprint

work page 2018

[4] [4]

Brendle, P.-K

S. Brendle, P.-K. Hung and M.-T. Wang, A Minkowski inequality for hypersurfaces in the Anti-de Sitter-Schwarzschild manifold , Communications on Pure and Applied Mathematics, 2016, 69(1): 124-144

work page 2016

[5] [5]

S.-Y. A. Chang and Y. Wang, Inequalities for quermassintegrals on k-convex domains, Advances in Mathematics, 2013, 248: 335-377

work page 2013

[6] [6]

S.-Y. A. Chang and Y. Wang, Some higher order isoperimetric inequalities via the method of optimal transport, International Mathematics Research Notices, 2014, 24: 6619-6644

work page 2014

[7] [7]

Chodosh, M

O. Chodosh, M. Eichmair and T. Koerber, On the Minkowski inequality near the sphere , arXiv preprint arXiv:2306.03848, 2023

work page arXiv 2023

[8] [8]

L. L. De Lima and F. Gir˜ ao, An Alexandrov-Fenchel-Type inequality in hyperbolic space with an application to a Penrose inequality , Annales Henri Poincar´ e, 2016,17(4): 979-1002

work page 2016

[9] [9]

Figalli, F

A. Figalli, F. Maggi and A. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities, Inventiones Mathematicae, 2010, 182(1): 167-211

work page 2010

[10] [10]

Fuglede, Stability in the isoperimetric problem for convex or nearly spherical domains in Rn, Transactions of the American Mathematical Society, 1989, 314(2): 619-638

B. Fuglede, Stability in the isoperimetric problem for convex or nearly spherical domains in Rn, Transactions of the American Mathematical Society, 1989, 314(2): 619-638

work page 1989

[11] [11]

Fusco, F

N. Fusco, F. Maggi and A. Pratelli, The sharp quantitative isoperimetric inequality , Annals of Mathematics, 2008, 168(3): 941-980

work page 2008

[12] [12]

Fusco, The quantitative isoperimetric inequality and related topics , Bulletin of Mathematical Sciences, 2015, 5(3): 517-607

N. Fusco, The quantitative isoperimetric inequality and related topics , Bulletin of Mathematical Sciences, 2015, 5(3): 517-607

work page 2015

[13] [13]

Fusco and D.A

N. Fusco and D.A. La Manna, Some weighted isoperimetric inequalities in quantitative form , Jour- nal of Functional Analysis, 2023, 285(2): 109946

work page 2023

[14] [14]

Gavitone, D.A

N. Gavitone, D.A. La Manna, G. Paoli, and L. Trani, A quantitative Weinstock inequality for convex sets, Calculus of Variations and Partial Differential Equations, 2020, 59(1): 2

work page 2020

[15] [15]

Glaudo, Minkowski inequality for nearly spherical domains , Advances in Mathematics, 2022, 408: 108595

F. Glaudo, Minkowski inequality for nearly spherical domains , Advances in Mathematics, 2022, 408: 108595. 29

work page 2022

[16] [16]

Gir˜ ao and N

F. Gir˜ ao and N. M. Pinheiro,An Alexandrov-Fenchel-type inequality for hypersurfaces in the sphere, Annals of Global Analysis and Geometry, 2017, 52(4): 413-424

work page 2017

[17] [17]

Gir˜ ao and D

F. Gir˜ ao and D. Rodrigues, Weighted geometric inequalities for hypersurfaces in sub-static mani- folds, Bulletin of the London Mathematical Society, 2020, 52(1): 121-136

work page 2020

[18] [18]

Guan and J

P. Guan and J. Li, The quermassintegral inequalities for k-convex starshaped domains , Advances in Mathematics, 2009, 221(5): 1725-1732

work page 2009

[19] [19]

Guan and J

P. Guan and J. Li, A mean curvature type flow in space forms, International Mathematics Research Notices, 2015, 13: 4716-4740

work page 2015

[20] [20]

Y. Ge, G. Wang and J. Wu, The GBC mass for asymptotically hyperbolic manifolds, Mathematische Zeitschrift, 2015, 281: 257-297

work page 2015

[21] [21]

Hu and H

Y. Hu and H. Li, Geometric inequalities for static convex domains in hyperbolic space, Transactions of the American Mathematical Society, 2022, 375(08): 5587-5615

work page 2022

[22] [22]

Y. Hu, H. Li and Y. Wei,Locally constrained curvature flows and geometric inequalities in hyperbolic space, Mathematische Annalen, 2022, 382(3-4): 1425-1474

work page 2022

[23] [23]

Kwong and P

K.-K. Kwong and P. Miao, A new monotone quantity along the inverse mean curvature flow in Rn, Pacific Journal of Mathematics, 2014, 267(2): 417-422

work page 2014

[24] [24]

Kwong and P

K.-K. Kwong and P. Miao, Monotone quantities involving a weighted σk integral along inverse curvature flows, Communications in Contemporary Mathematics, 2015, 17(05): 1550014

work page 2015

[25] [25]

Kwong and Y

K.-K. Kwong and Y. Wei, Geometric inequalities involving three quantities in warped product man- ifolds, Advances in Mathematics, 2023, 430: 109213

work page 2023

[26] [26]

Alexandrov-Fenchel type inequalities with convex weight in space forms

K.-K. Kwong and Y. Wei, Alexandrov-Fenchel type inequalities with convex weight in space forms , arXiv preprint arXiv:2412.08923, 2024

work page internal anchor Pith review Pith/arXiv arXiv 2024

[27] [27]

Kwong, Y

K.-K. Kwong, Y. Wei, G. Wheeler, and V.-M. Wheeler, On an inverse curvature flow in two- dimensional space forms, Mathematische Annalen, 2022, 384(1): 1-24

work page 2022

[28] [28]

Maggi, Some methods for studying stability in isoperimetric type problems, Bulletin of the Amer- ican Mathematical Society, 2008, 45(3): 367-408

F. Maggi, Some methods for studying stability in isoperimetric type problems, Bulletin of the Amer- ican Mathematical Society, 2008, 45(3): 367-408

work page 2008

[29] [29]

Pan and J

P. Pan and J. Liu, New weighted Alexandrov-Fenchel type inequalities for hypersurfaces in hyperbolic space, Journal of Geometry and Physics, 2025: 105528

work page 2025

[30] [30]

Qiu, A family of higher-order isoperimetric inequalities , Communications in Contemporary Mathematics, 2015, 17(03): 1450015

G. Qiu, A family of higher-order isoperimetric inequalities , Communications in Contemporary Mathematics, 2015, 17(03): 1450015

work page 2015

[31] [31]

Solanes, Integral geometry and the Gauss-Bonnet theorem in constant curvature spaces , Trans- actions of the American Mathematical Society, 2006, 358(3): 1105-1115

G. Solanes, Integral geometry and the Gauss-Bonnet theorem in constant curvature spaces , Trans- actions of the American Mathematical Society, 2006, 358(3): 1105-1115

work page 2006

[32] [32]

Scheuer and C

J. Scheuer and C. Xia, Locally constrained inverse curvature flows , Transactions of the American Mathematical Society, 2019, 372(10): 6771-6803

work page 2019

[33] [33]

VanBlargan and Y

C. VanBlargan and Y. Wang, Quantitative quermassintegral inequalities for nearly spherical sets , Communications in Contemporary Mathematics, 2024, 26(06): 2350026

work page 2024

[34] [34]

Wei and T

Y. Wei and T. Zhou, New weighted geometric inequalities for hypersurfaces in space forms, Bulletin of the London Mathematical Society, 2023, 55(1): 263-281

work page 2023

[35] [35]

Wu, Weighted Alexandrov-Fenchel type inequalities for hypersurfaces in Rn, Bulletin of the London Mathematical Society, 2024, 56(8): 2634-2646

J. Wu, Weighted Alexandrov-Fenchel type inequalities for hypersurfaces in Rn, Bulletin of the London Mathematical Society, 2024, 56(8): 2634-2646

work page 2024

[36] [36]

Wu, New weighted Alexandrov-Fenchel type inequalities and Minkowski inequalities in space forms, arXiv:2506.22877, 2025

J. Wu, New weighted Alexandrov-Fenchel type inequalities and Minkowski inequalities in space forms, arXiv:2506.22877, 2025

work page arXiv 2025

[37] [37]

Zhou and T

R. Zhou and T. Zhou, Stability of Alexandrov–Fenchel type inequalities for nearly spherical sets in space forms, The Journal of Geometric Analysis, 2024, 34(12): 376. 30 Weimin Sheng: School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China. Email address : shengweimin@zju.edu.cn Yinhang W ang: School of Mathematical Sciences, Zhejiang...

work page 2024