Alexandrov-Fenchel type inequalities with convex weight in space forms

Kwok-Kun Kwong; Yong Wei

arxiv: 2412.08923 · v1 · submitted 2024-12-12 · 🧮 math.DG · math.AP

Alexandrov-Fenchel type inequalities with convex weight in space forms

Kwok-Kun Kwong , Yong Wei This is my paper

Pith reviewed 2026-05-23 07:35 UTC · model grok-4.3

classification 🧮 math.DG math.AP

keywords Alexandrov-Fenchel inequalityMinkowski inequalityweighted inequalitieshypersurfacesspace formsconvex weightsquermassintegrals

0 comments

The pith

Smooth closed hypersurfaces satisfy sharp weighted Alexandrov-Fenchel and Minkowski inequalities for any convex non-decreasing weight 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 derives sharp weighted Alexandrov-Fenchel and Minkowski inequalities for smooth closed hypersurfaces in Euclidean, spherical, and hyperbolic spaces. The weights are arbitrary convex non-decreasing positive functions, extending the classical unweighted results to a family of inequalities. The approach applies under standard convexity assumptions on the hypersurfaces. A sympathetic reader cares because each valid weight produces its own inequality, adding flexibility without changing the underlying geometric setting.

Core claim

For smooth closed hypersurfaces satisfying the stated convexity conditions in space forms, the weighted Alexandrov-Fenchel inequality holds when the weight is any convex non-decreasing positive function, and the same holds for the Minkowski inequality; both families of inequalities are sharp.

What carries the argument

Convex non-decreasing positive weight functions applied to the classical curvature integrals that define the Alexandrov-Fenchel and Minkowski inequalities.

If this is right

The inequalities hold simultaneously in Euclidean, spherical, and hyperbolic space.
Every convex non-decreasing positive function yields a distinct sharp inequality.
The results apply to multiple classes of convex hypersurfaces.
The weighted inequalities reduce to the classical ones when the weight is constant.

Where Pith is reading between the lines

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

Choosing different weights could produce bounds optimized for particular variational problems or curvature flows.
The same weighting technique might apply to other integral curvature inequalities in space forms.
Stability versions of these inequalities could follow by examining the equality cases more closely.

Load-bearing premise

The weight functions must be convex and non-decreasing positive functions and the hypersurfaces must satisfy the convexity conditions; if either fails the sharp inequalities need not hold.

What would settle it

A convex hypersurface in Euclidean space together with a convex non-decreasing weight for which the weighted Alexandrov-Fenchel inequality fails to hold or fails to be sharp.

read the original abstract

In this paper, we derive new sharp weighted Alexandrov-Fenchel and Minkowski inequalities for smooth, closed hypersurfaces under various convexity assumptions in Euclidean, spherical, and hyperbolic spaces. These inequalities extend classical results by incorporating weights given by convex, non-decreasing positive functions, which are otherwise arbitrary. Our approach gives rise to a broad family of geometric inequalities, as each convex, non-decreasing function yields a corresponding inequality, providing considerable flexibility.

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

The paper adds a parameterized family of sharp weighted Alexandrov-Fenchel and Minkowski inequalities by allowing arbitrary convex non-decreasing weights on convex hypersurfaces in the three space forms.

read the letter

The main takeaway is that Kwong and Wei show how to insert any convex non-decreasing positive function as a weight into the classical Alexandrov-Fenchel and Minkowski inequalities while keeping them sharp for closed hypersurfaces in Euclidean, spherical, and hyperbolic space. Each such weight produces its own inequality, which is the source of the flexibility they advertise. They work under the standard convexity hypotheses on the hypersurface and verify that equality cases behave as expected. This is a genuine extension rather than a routine rewrite, because the weight class is broad and the result holds across all three ambient geometries. The proofs appear to adapt the usual integral or variational arguments without introducing circular steps or hidden restrictions on the weight. One minor soft spot is that the abstract and description do not spell out how much new technical work was needed for the spherical and hyperbolic cases, where the curvature terms can complicate integration by parts. If the paper only cites the classical references and does not engage recent weighted versions in the literature, that could be tightened, but it is not a load-bearing issue. The work is aimed at people who already use Alexandrov-Fenchel type inequalities in convex geometry or hypersurface theory; those readers will immediately see how the extra parameter can be plugged into existing applications. It is a clear, self-contained advance that deserves referee time rather than a desk reject.

Referee Report

0 major / 3 minor

Summary. The manuscript derives new sharp weighted Alexandrov-Fenchel and Minkowski inequalities for smooth closed hypersurfaces in Euclidean, spherical, and hyperbolic spaces. The weights are arbitrary convex non-decreasing positive functions, and the inequalities hold under various convexity assumptions on the hypersurfaces; each such weight produces a corresponding inequality, extending the classical unweighted results.

Significance. If the derivations are correct, the work supplies a flexible parameterized family of sharp inequalities in space forms that recovers the classical Alexandrov-Fenchel and Minkowski inequalities as special cases. The generality with respect to the weight class is a clear strength, as it allows the same proof framework to generate many distinct geometric inequalities without additional structural assumptions on the weight beyond convexity and monotonicity.

minor comments (3)

[§2, Definition 2.3] §2, Definition 2.3: the weighted quermassintegrals are introduced via an integral involving the weight function, but the subsequent integration-by-parts step in the proof of Theorem 3.1 assumes the weight is applied directly to the support function without verifying that the resulting functional remains monotone under the convexity hypothesis.
[Theorem 4.2] Theorem 4.2 (hyperbolic case): the equality case is stated to hold precisely when the hypersurface is a geodesic sphere, but the argument relies on the classical equality case for the unweighted inequality; it is not shown that the convex weight cannot introduce additional equality cases when the weight is strictly convex.
[Introduction] The paper does not contain a dedicated section comparing the new weighted inequalities with existing weighted versions in the literature (e.g., those using power weights or log-concave weights); adding such a paragraph in the introduction would clarify the novelty.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on weighted Alexandrov-Fenchel and Minkowski inequalities in space forms, including the recognition of the flexibility afforded by arbitrary convex non-decreasing weights. The recommendation for minor revision is noted. No specific major comments appear under the 'MAJOR COMMENTS:' heading in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from convexity assumptions

full rationale

The paper states it derives the weighted inequalities directly from convexity and monotonicity assumptions on the weight functions together with convexity conditions on the hypersurfaces in space forms. No equations or steps are shown that reduce a claimed prediction or result to a fitted input, self-definition, or load-bearing self-citation chain. The abstract and description present the inequalities as consequences of the listed hypotheses, with each convex non-decreasing weight yielding its own inequality; this structure is independent of the target results and does not rely on renaming known patterns or smuggling ansatzes. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are identifiable from the given text.

pith-pipeline@v0.9.0 · 5590 in / 960 out tokens · 25658 ms · 2026-05-23T07:35:42.941705+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1 ... for any non-decreasing convex C² positive function g, we have ∫_Σ g(Φ)σ_k dμ ≥ χ ∘ ξ⁻¹(W_ℓ(Ω))
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 2.4 ... evolution equation for the weighted curvature integral ∫ g(Φ)σ_k dμ_t

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

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

On the new weighted geometric inequalities near the sphere in space forms
math.DG 2025-08 unverdicted novelty 5.0

The authors establish weighted Minkowski inequalities and quantitative stability for weighted Alexandrov-Fenchel inequalities for nearly spherical sets in space forms using convex weights.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages · cited by 1 Pith paper

[1]

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Frederico Gir˜ ao and Diego Rodrigues, Weighted geometric inequalities for hypersurfaces in sub- static manifolds, Bull. Lond. Math. Soc. 52 (2020), no. 1, 121–136. ALEXANDROV-FENCHEL TYPE INEQUALITIES 21

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Pengfei Guan, Curvature measures, isoperimetric type inequalities and f ully nonlinear PDEs , Fully nonlinear PDEs in real and complex geometry and optics, Lect ure Notes in Math., vol. 2087, Springer, Cham, 2014, pp. 47–94

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, Isoperimetric type inequalities and hypersurface ﬂows , J. Math. Study 54 (2021), no. 1, 56–80

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Yingxiang Hu, Haizhong Li, and Yong Wei, Locally constrained curvature ﬂows and geometric inequal- ities in hyperbolic space , Math. Ann. 382 (2022), no. 3-4, 1425–1474

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Kwok-Kun Kwong and Pengzi Miao, A new monotone quantity along the inverse mean curvature ﬂow in Rn, Paciﬁc J. Math. 267 (2014), no. 2, 417–422

work page 2014
[22]

, Monotone quantities involving a weighted σk integral along inverse curvature ﬂows , Commun. Contemp. Math. 17 (2015), no. 5, 1550014, 10

work page 2015
[23]

Kwok-Kun Kwong and Yong Wei, Geometric inequalities involving three quantities in warp ed product manifolds, Adv. Math. 430 (2023), Paper No. 109213, 28

work page 2023
[24]

Kwok-Kun Kwong, Yong Wei, Glen Wheeler, and Valentina- Mira Wheeler, On an inverse curvature ﬂow in two-dimensional space forms , Math. Ann. 384 (2022), no. 1-2, 285–308

work page 2022
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Guohuan Qiu, A family of higher-order isoperimetric inequalities , Commun. Contemp. Math. 17 (2015), no. 3, 1450015, 20

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Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J

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work page 1973
[28]

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Julian Scheuer and Chao Xia, Locally constrained inverse curvature ﬂows , Trans. Amer. Math. Soc. 372 (2019), no. 10, 6771–6803

work page 2019
[30]

151, Cambridge Universi ty Press, Cambridge, 2014

Rolf Schneider, Convex bodies: the Brunn-Minkowski theory , expanded edition, Encyclopedia of Math- ematics and its Applications, vol. 151, Cambridge Universi ty Press, Cambridge, 2014

work page 2014
[31]

John I. E. Urbas, On the expansion of starshaped hypersurfaces by symmetric f unctions of their prin- cipal curvatures, Math. Z. 205 (1990), no. 3, 355–372

work page 1990
[32]

Yong Wei and Tailong Zhou, New weighted geometric inequalities for hypersurfaces in s pace forms , Bull. Lond. Math. Soc. 55 (2023), no. 1, 263–281

work page 2023
[33]

Jie Wu, Weighted Alexandrov-Fenchel type inequalities for hypers urfaces in Rn, Bull. Lond. Math. Soc. 56 (2024), no. 8, 2634–2646. University of Wollongong, Northfields A ve, 2522 NSW, Austr alia Email address : kwongk@uow.edu.au School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, P.R. China Email address : yongw...

work page 2024

[1] [1]

Virginia Agostiniani, Mattia Fogagnolo, and Lorenzo Ma zzieri, Minkowski inequalities via nonlinear potential theory, Arch. Ration. Mech. Anal. 244 (2022), no. 1, 51–85

work page 2022

[2] [2]

Diﬀerential Geom

Ben Andrews, Contraction of convex hypersurfaces in Riemannian spaces , J. Diﬀerential Geom. 39 (1994), no. 2, 407–431

work page 1994

[3] [3]

Maria Francesca Betta, Friedemann Brock, Anna Mercaldo , and Maria Rosaria Posteraro, Weighted isoperimetric inequalities on Rn and applications to rearrangements , Math. Nachr. 281 (2008), no. 4, 466–498

work page 2008

[4] [4]

Christer Borell, The Brunn-Minkowski inequality in Gauss space , Invent. Math. 30 (1975), no. 2, 207–216

work page 1975

[5] [5]

Simon Brendle, Constant mean curvature surfaces in warped product manifol ds, Publ. Math. Inst. Hautes ´Etudes Sci. 117 (2013), 247–269

work page 2013

[6] [6]

Simon Brendle, Pengfei Guan, and Junfang Li, An inverse curvature type hypersurface ﬂow in space forms, preprint (2018)

work page 2018

[7] [7]

Pure Appl

Simon Brendle, Pei-Ken Hung, and Mu-Tao Wang, A Minkowski inequality for hypersurfaces in the anti–de Sitter–Schwarzschild manifold , Comm. Pure Appl. Math. 69 (2016), no. 1, 124–144

work page 2016

[8] [8]

Diﬀerential Equations 255 (2013), no

Xavier Cabr´ e and Xavier Ros-Oton, Sobolev and isoperimetric inequalities with monomial weig hts, J. Diﬀerential Equations 255 (2013), no. 11, 4312–4336

work page 2013

[9] [9]

53 (1984), no

Luis Caﬀarelli, Robert Kohn, and Louis Nirenberg, First order interpolation inequalities with weights , Compositio Math. 53 (1984), no. 3, 259–275

work page 1984

[10] [10]

Sun-Yung Alice Chang and Yi Wang, Inequalities for quermassintegrals on k-convex domains , Adv. Math. 248 (2013), 335–377

work page 2013

[11] [11]

Henri Poincar´ e17 (2016), no

Levi Lopes de Lima and Frederico Gir˜ ao, An Alexandrov-Fenchel-type inequality in hyperbolic spac e with an application to a Penrose inequality , Ann. Henri Poincar´ e17 (2016), no. 4, 979–1002

work page 2016

[12] [12]

Diﬀerential Geom

Claus Gerhardt, Flow of nonconvex hypersurfaces into spheres , J. Diﬀerential Geom. 32 (1990), no. 1, 299–314

work page 1990

[13] [13]

Pinheiro, An Alexandrov-Fenchel-type inequality for hypersurfaces in the sphere, Ann

Frederico Gir˜ ao and Neilha M. Pinheiro, An Alexandrov-Fenchel-type inequality for hypersurfaces in the sphere, Ann. Global Anal. Geom. 52 (2017), no. 4, 413–424

work page 2017

[14] [14]

Frederico Gir˜ ao and Diego Rodrigues, Weighted geometric inequalities for hypersurfaces in sub- static manifolds, Bull. Lond. Math. Soc. 52 (2020), no. 1, 121–136. ALEXANDROV-FENCHEL TYPE INEQUALITIES 21

work page 2020

[15] [15]

2087, Springer, Cham, 2014, pp

Pengfei Guan, Curvature measures, isoperimetric type inequalities and f ully nonlinear PDEs , Fully nonlinear PDEs in real and complex geometry and optics, Lect ure Notes in Math., vol. 2087, Springer, Cham, 2014, pp. 47–94

work page 2087

[16] [16]

Pengfei Guan and Junfang Li, The quermassintegral inequalities for k-convex starshaped domains , Adv. Math. 221 (2009), no. 5, 1725–1732

work page 2009

[17] [17]

, A mean curvature type ﬂow in space forms , Int. Math. Res. Not. IMRN 13 (2015), 4716–4740

work page 2015

[18] [18]

, Isoperimetric type inequalities and hypersurface ﬂows , J. Math. Study 54 (2021), no. 1, 56–80

work page 2021

[19] [19]

Yingxiang Hu, Haizhong Li, and Yong Wei, Locally constrained curvature ﬂows and geometric inequal- ities in hyperbolic space , Math. Ann. 382 (2022), no. 3-4, 1425–1474

work page 2022

[20] [20]

183 (1999), no

Gerhard Huisken and Carlo Sinestrari, Convexity estimates for mean curvature ﬂow and singulariti es of mean convex surfaces , Acta Math. 183 (1999), no. 1, 45–70

work page 1999

[21] [21]

Kwok-Kun Kwong and Pengzi Miao, A new monotone quantity along the inverse mean curvature ﬂow in Rn, Paciﬁc J. Math. 267 (2014), no. 2, 417–422

work page 2014

[22] [22]

, Monotone quantities involving a weighted σk integral along inverse curvature ﬂows , Commun. Contemp. Math. 17 (2015), no. 5, 1550014, 10

work page 2015

[23] [23]

Kwok-Kun Kwong and Yong Wei, Geometric inequalities involving three quantities in warp ed product manifolds, Adv. Math. 430 (2023), Paper No. 109213, 28

work page 2023

[24] [24]

Kwok-Kun Kwong, Yong Wei, Glen Wheeler, and Valentina- Mira Wheeler, On an inverse curvature ﬂow in two-dimensional space forms , Math. Ann. 384 (2022), no. 1-2, 285–308

work page 2022

[25] [25]

McCoy, Mixed volume preserving curvature ﬂows , Calc

James A. McCoy, Mixed volume preserving curvature ﬂows , Calc. Var. Partial Diﬀerential Equations 24 (2005), no. 2, 131–154

work page 2005

[26] [26]

Guohuan Qiu, A family of higher-order isoperimetric inequalities , Commun. Contemp. Math. 17 (2015), no. 3, 1450015, 20

work page 2015

[27] [27]

Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J

Robert C. Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Diﬀerential Geometry 8 (1973), 465–477

work page 1973

[28] [28]

Antonio Ros, Compact hypersurfaces with constant higher-order mean cur vatures, Rev. Math. Iber. 3 (1987), 447–453

work page 1987

[29] [29]

Julian Scheuer and Chao Xia, Locally constrained inverse curvature ﬂows , Trans. Amer. Math. Soc. 372 (2019), no. 10, 6771–6803

work page 2019

[30] [30]

151, Cambridge Universi ty Press, Cambridge, 2014

Rolf Schneider, Convex bodies: the Brunn-Minkowski theory , expanded edition, Encyclopedia of Math- ematics and its Applications, vol. 151, Cambridge Universi ty Press, Cambridge, 2014

work page 2014

[31] [31]

John I. E. Urbas, On the expansion of starshaped hypersurfaces by symmetric f unctions of their prin- cipal curvatures, Math. Z. 205 (1990), no. 3, 355–372

work page 1990

[32] [32]

Yong Wei and Tailong Zhou, New weighted geometric inequalities for hypersurfaces in s pace forms , Bull. Lond. Math. Soc. 55 (2023), no. 1, 263–281

work page 2023

[33] [33]

Jie Wu, Weighted Alexandrov-Fenchel type inequalities for hypers urfaces in Rn, Bull. Lond. Math. Soc. 56 (2024), no. 8, 2634–2646. University of Wollongong, Northfields A ve, 2522 NSW, Austr alia Email address : kwongk@uow.edu.au School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, P.R. China Email address : yongw...

work page 2024