Affine isoperimetric type inequalities for static convex domains in hyperbolic space

Botong Xu; Haizhong Li; Yao Wan; Yingxiang Hu

REVIEW 1 major objections 4 minor 40 references

Affine isoperimetric type inequalities for static convex domains in hyperbolic space

T0 review · 1 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For static convex domains in hyperbolic space, sharp affine isoperimetric inequalities hold with equality precisely when the domain is a hyperbolic ellipsoid.

desk verdict The projection identities and Theorem 1.1 are solid, but Proposition 2.5 is false and it is load-bearing for Theorems 1.2 and 1.3, so those two results are unproved as written. read the letter →

arxiv 2504.15677 v1 pith:IXDC46LC submitted 2025-04-22 math.DG

Yingxiang Hu , Haizhong Li , Yao Wan , Botong Xu This is my paper · ORCID

classification math.DG MSC 52A4053C2453A15

keywords staticconvexityhyperbolicellipsoidaffineisoperimetricinequalityBlaschke-SantalóorthogonalprojectionspaceLp-affinesurfacearea

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

This paper proves a family of sharp affine isoperimetric inequalities for smooth bounded domains in hyperbolic space whose boundaries are static convex with respect to a fixed point. The main theorem bounds an integral of a normalized curvature on the boundary by a power of the weighted volume, with equality exactly for the newly introduced hyperbolic ellipsoids. Versions for Lp-affine surface area and for the hyperbolic polar body also follow. The proof works by orthogonally projecting the hyperbolic domain onto Euclidean space and showing that the hyperbolic affine surface area equals the Euclidean affine surface area of the projection, so classical Euclidean theorems apply directly.

What carries the argument

The load-bearing mechanism is the orthogonal (Gibbons) projection $\pi_{p_0}: H^n \to \mathbb{R}^n$ with respect to a point $p_0$, which sends a static convex domain $K$ to a Euclidean convex body $\hat K = \pi_{p_0}(K)$. The key identity is $\operatorname{as}^H_p(K) = \operatorname{as}_p(\hat K)$, obtained by combining the volume-element identity $V\,d\mathrm{vol} = d\mathrm{vol}_{\hat K}$, the support-function relation $u = ((1+|x|^2)/(1+\hat u^2))^{1/2}\hat u$, and the Gauss-curvature transformation $H_{n-1}(\tilde{\kappa}) = ((1+|x|^2)/(1+\hat u^2))^{(n+1)/2}H_{n-1}(\hat{\kappa})$, where $\tilde{\kappa}_i = V\kappa_i - V_{,\nu}$. Static convexity of $\partial K$ is equivalent to Euclidean convexity of $\partial\hat K$, so the Euclidean affine isoperimetric theorems apply directly, and equality in the hyperbolic inequalities is governed by the projection being an ellipsoid.

What would settle it

Project a geodesic ball in $H^n$ centered away from $p_0$ (which is not a geodesic ball centered at $p_0$) and compute both sides of (1.7); the identity should hold exactly because the projection is an ellipsoid. Then perform the same computation for a static convex domain whose projection is a smoothly rounded cube: the inequality should be strict, and any equality would disprove the equality characterization.

Watch

Extended reading notes

Core claim

The central discovery is that the classical affine isoperimetric inequalities of Euclidean convex geometry transfer to static convex domains in hyperbolic space, with equality characterized by a new class of extremizers: hyperbolic ellipsoids, defined as the inverse projection of Euclidean ellipsoids. The main theorem states that if $K$ is a smooth bounded domain in $H^n$ whose boundary is static convex with respect to a point $p_0$, then $\int_{\partial K} H_{n-1}(\tilde{\kappa})^{1/(n+1)}\,dA \le n|B^n|^{2/(n+1)} (\int_K V\,d\mathrm{vol})^{(n-1)/(n+1)}$, with equality if and only if $K$ is a hyperbolic ellipsoid. The paper also proves Lp-versions (Theorem 1.2) and a Blaschke-Santal\'o type inequality for the hyperbolic polar body (Theorem 1.3). All of these follow from a single mechanism: orthogonal projection onto Euclidean space, under which static convexity becomes ordinary convexity and the hyperbolic affine surface area equals the Euclidean affine surface area of the projected body.

Load-bearing premise

The load-bearing premise is the exact identity $\operatorname{as}^H_p(K)=\operatorname{as}_p(\pi_{p_0}(K))$ between hyperbolic and Euclidean Lp-affine surface areas for static convex domains; if the support-function, Gauss-curvature, or volume-element transformation formulas connecting $H^n$ to its projection had different exponents or signs, the stated inequalities would not follow.

Editorial extensions

If this is right

Theorem 1.1 gives a sharp curvature-integral bound for static convex domains in hyperbolic space: no such domain can beat the weighted-volume power law, and the only equality cases are hyperbolic ellipsoids.
Theorem 1.2 transplants Lutwak's Lp-affine isoperimetric theory into hyperbolic space, preserving the three regimes $p>0$, $-n<p<0$, and $p<-n$, with equality for hyperbolic ellipsoids centered at the hyperbolic centroid.
Theorem 1.3 yields a sharp Blaschke-Santal\'o inequality for the hyperbolic polar body, bounding the product of weighted volumes by $|B^n|^2$ with equality exactly for centered hyperbolic ellipsoids.
Corollary 2.3 provides a dictionary: static convexity in $H^n$ is synonymous with convexity of the Gibbons projection, so any affine isoperimetric theorem for Euclidean convex bodies that is $\mathrm{SL}(n)$-invariant automatically produces a hyperbolic analogue.
Remark 1.4 indicates the same projection proof works in the unit sphere with $V = \cos r$, so the paper's inequalities hold for static convex domains in spherical space as well.

Reading between the lines

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

The projection identity is strong enough to transfer not only the inequalities but the whole structural theory: for instance, the duality $\operatorname{as}_p(K)=\operatorname{as}_{n^2/p}(K^\circ)$ from Proposition 3.4 suggests that hyperbolic static convex domains form an Lp-affine theory parallel to Euclidean convex bodies, with polar duality preserved under projection.
Since the extremizers are hyperbolic ellipsoids, one expects a rigidity phenomenon: any static convex domain that saturates one of the inequalities must have a Euclidean ellipsoid as its projection, which could be checked numerically for non-ellipsoidal convex bodies.
The non-sharp constant in (1.10) comes from the inverse Santal\'o inequality; tightening the Euclidean constant in that regime would immediately tighten the hyperbolic inequality, a path the paper does not pursue.
The same projection technique may extend beyond constant-curvature space forms to static spacetimes, where static convexity was originally defined, potentially linking affine isoperimetric inequalities to quasi-local mass.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know. The projection identities (Lemmas 2.1–2.2) are correct, and the reduction as_H^p(K)=as_p(\hat K) in (3.12) checks out term by term. Theorem 1.1 is a genuine and clean new hyperbolic affine isoperimetric inequality. But the paper’s other load-bearing step, Proposition 2.5, is false, and it is needed for Theorems 1.2 and 1.3. The stress-test note missed this.

What is new and good: the hyperbolic ellipsoid definition, the hyperbolic polar body, and the idea of transferring the Euclidean affine theory through Gibbons projection. The algebra in Section 3 is careful, and the equality discussion for Theorem 1.1 is sound.

The soft spot: Proposition 2.5 claims that the hyperbolic centroid of K is at p_0 if and only if the Euclidean centroid of \hat K = π_{p0}(K) is at the origin. From the definition cen(K)=∫X dvol /‖∫X dvol‖, the condition for the centroid to be at the base point is ∫_{\hat K} x (1/V) dA = 0 (with dA the Euclidean area), not ∫_{\hat K} x dA = 0. These differ by the positive factor 1/V=1/√(1+|x|^2). A concrete convex body in R^2 with vertices (-1,-1), (-1,1), (2,0) has Euclidean centroid at the origin, but ∫ x/V dA is strictly negative. So the equivalence is false. The proof of Proposition 2.5 computes the gradient of ∫ cosh d(Y,X) dvol incorrectly, picking up an extra cosh r factor.

This matters because Theorems 1.2 and 1.3 invoke Theorems D and E, which require the Euclidean centroid at the origin. With Proposition 2.5 false, those reductions collapse. Proposition 3.2’s second sentence (the polar of a centered body is centered) is also false, but it is not used in the main proofs.

So: Theorem 1.1 stands; Theorems 1.2 and 1.3 are unproved as written. I would send this to a referee because the core framework is worth engaging and the error is fixable (e.g., replace the hyperbolic centroid assumption with the Euclidean-centroid-of-projection condition, or prove a different relation). But the authors should be told clearly that two of the three advertised results lack proof.

Send for review, but expect a major revision.

Referee Report

1 major / 4 minor

Summary. The paper introduces the notion of hyperbolic ellipsoids as preimages, under a natural orthogonal projection, of Euclidean ellipsoids, and also defines hyperbolic centroids and hyperbolic polar bodies. The main results, Theorems 1.1–1.3, are sharp affine isoperimetric-type inequalities for static convex domains in hyperbolic space: an affine surface area bound, Lp-affine surface area bounds, and a Blaschke–Santaló type weighted volume product inequality. The proofs use the hyperboloid model, define the orthogonal projection π_{p0}, and derive in Lemmas 2.1–2.2 explicit formulas relating the metric, normal, support function, second fundamental form, volume/area elements, and Gauss curvature of a hypersurface and its projection. These identities yield equation (3.12), the equality of hyperbolic and Euclidean Lp-affine surface areas, after which the theorems follow by applying Petty's affine isoperimetric inequality, the Lutwak–Werner–Ye Lp inequalities, and the Blaschke–Santaló inequality. Equality is characterized by hyperbolic ellipsoids.

Significance. If correct, the paper establishes a clean and useful bridge between hyperbolic static-convex geometry and Euclidean convex geometry, yielding sharp inequalities with explicit constants (except for the non-sharp c in (1.10)). The central reduction (3.12) is a genuine parameter-free identity rather than an estimate, and Lemmas 2.1–2.2 are derived carefully. The paper's notions of hyperbolic ellipsoid, hyperbolic centroid, and hyperbolic polar body are natural and likely to be reusable. I find no circularity, no hidden fitting, and no unsupported assumption in the main inequality proofs; the algebraic bridge is sound.

major comments (1)

[Section 1, Theorem 1.1 (and Theorem 1.2)] The equality statement 'K is a hyperbolic ellipsoid' is ambiguous because the definition of hyperbolic ellipsoid is existential: it requires that the projection be an ellipsoid for some point. The proof of Theorem 1.1 in Section 4 establishes equality in (1.7) if and only if \hat K = π_{p0}(K) is an ellipsoid, i.e., if and only if K is a hyperbolic ellipsoid with respect to the specific point p0 appearing in the hypothesis. As written, the 'if' direction is not established for a hyperbolic ellipsoid defined with respect to a different base point, and the property is base-point dependent in general. Please restate the equality case as 'with respect to p0' and similarly in Theorem 1.2, where the equality condition should refer to the hyperbolic centroid p0.

minor comments (4)

[Section 3, Proposition 3.2] The second sentence of Proposition 3.2, claiming that K° is static convex with respect to its hyperbolic centroid p0 whenever K is, is false: polar duality in Euclidean space does not preserve centroids, and a generic asymmetric convex body with centroid at the origin has a polar whose centroid is not the origin. This statement is not used in the proofs of Theorems 1.1–1.3—only the identity π_{p0}(K°) = (π_{p0}(K))° is used—so the main results are unaffected, but the proposition should be corrected or weakened to static convexity with respect to the original p0.
[Section 3, paragraph after Theorem D] The identity 'as_0(\hat K) = 1/n Vol(\hat K)' appears inconsistent with definition (3.3); substituting p = 0 gives ∫ \hat u d\hat A = n Vol(\hat K). Please check and correct this formula.
[Section 3, proof of Proposition 3.2] The typo 'cosd' should read 'cosh' in the displayed formulas of the proof of Proposition 3.2.
[Section 3, after (3.12)] The notation 'asp(K)' without the superscript H should be 'asH_p(K)' for consistency with the definition (3.10).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the hyperbolic inequalities are obtained by explicit projection identities from external Euclidean affine isoperimetric theorems.

full rationale

The central reduction is as_H^p(K) = as_p(pi_{p0}(K)) at equation (3.12), built from the explicit formulas (2.6), (2.11), and (2.13); no fitted parameter and no assumed target inequality enters. Theorems 1.1-1.3 then follow by applying the external Euclidean affine isoperimetric inequality (Theorem C), the L^p-affine inequalities (Theorem D), and the Blaschke-Santalo inequality (Theorem E) to the convex body bK = pi_{p0}(K). Equality characterizations transfer because 'hyperbolic ellipsoid' is defined as the preimage of a Euclidean ellipsoid, so the equality case is a translation of the Euclidean equality case, not a circular assumption. The only self-citations, [24, Lem. 2.1 and 2.3] for formulas (2.10)-(2.11), supply elementary, parameter-free, independently checkable projection identities; they do not assume the target results and are not load-bearing in a circular sense. The erroneous second sentence of Proposition 3.2, asserting that K^circ is static convex with respect to its hyperbolic centroid, is a genuine correctness defect in a non-central statement, but it is not used in the proofs of Theorems 1.1-1.3 and does not create circularity. Overall, the derivation chain is self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

The central claim is a geometric inequality; the terms in the inequality are defined through standard hyperbolic geometry (hyperboloid model, static convexity) and the proof uses external Euclidean affine isoperimetric theorems as black boxes. No numerical parameters are fitted; the only constants are |B^n| and the universal Bourgain-Milman constant c.

assumptions (5)

standard math Classical Euclidean affine isoperimetric inequalities (Theorems C, D, E) hold for convex bodies in R^n.
Used as black boxes in the proofs of Theorems 1.1-1.3. External results.
domain assumption The orthogonal projection π_{p0} from the hyperboloid model satisfies the volume identity V dvol = d vol̂ and curvature identities (2.10)-(2.13).
Derived in Lemmas 2.1-2.2 using the hyperboloid model; some parts cited from [24].
domain assumption Static convexity of ∂K with respect to p0 is equivalent to convexity of the projected boundary in R^n.
Corollary 2.3, derived from (2.12).
domain assumption The hyperbolic centroid minimizes ∫ cosh d(Y,X) dvol, and its projection is the Euclidean centroid.
Propositions 2.4 and 2.5.
standard math Inverse Santaló inequality (Bourgain-Milman) provides the non-sharp constant c in (1.10).
Used in Theorem 1.2(iii) via Theorem D(iii).

invented entities (3)

Hyperbolic ellipsoid
purpose: Equality case in Theorems 1.1-1.3; defined as the preimage of a Euclidean ellipsoid under the Gibbons projection.
Definitional object with no external falsifiable prediction; its significance comes from the sharpness characterization inside the paper.
Hyperbolic polar body K°
purpose: Extends the Euclidean polar body to hyperbolic space for the Blaschke-Santaló analog (Theorem 1.3).
Defined by (1.11); duality properties are proven internally.
Hyperbolic centroid
purpose: Gives a canonical base point for the inequalities and ensures the projected body has Euclidean centroid at the origin.
New notion defined by (2.15); minimized ∫ cosh d(Y,X) dvol, proven in Proposition 2.4.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Affine isoperimetric type inequalities for static convex domains in hyperbolic space." pith.science (2026). https://pith.science/paper/IXDC46LC

@misc{pith2026250415677,
  author       = {Pith},
  title        = {Pith review of: Affine isoperimetric type inequalities for static convex domains in hyperbolic space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IXDC46LC}},
  note         = {Machine review of arXiv:2504.15677}
}

read the original abstract

In this paper, the notion of hyperbolic ellipsoids in hyperbolic space is introduced. Using a natural orthogonal projection from hyperbolic space to Euclidean space, we establish affine isoperimetric type inequalities for static convex domains in hyperbolic space. Moreover, equality of such inequalities is characterized by these hyperbolic ellipsoids.

Figures

Figures reproduced from arXiv: 2504.15677 by the authors.

**Figure 2.1.** Orthogonal projection π translation Tp0 ∈ Isom(Hn ) which sends p0 to N = (1, 0). Then the orthogonal projection with respect to this point p0 is given by πp0 = π ◦ Tp0 . (2.2) For any bounded domain K with smooth boundary ∂K in Hn , the orthogonal projection Kb = πp0 (K) with respect to a point p0 ∈ Hn is a smooth bounded domain in R n . This projection was first proposed by Gibbons [15] to prove the Penrose inequa… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

40 extracted references · 39 canonical work pages

[1]

Agostiniani, M

V. Agostiniani, M. Fogagnolo, and L. Mazzieri, Minkowski Inequalities via Nonlinear Potential Theory, Archive for Rational Mechanics and Analysis 244 (2022), 51–85

work page 2022
[2]

Andrews, X

B. Andrews, X. Chen, and Y. Wei, Volume preserving flow and Alexandrov-Fenchel type inequalities in hyperbolic space, J. Eur. Math. Soc.(JEMS) 23 (2021), no. 7, 2467–2509

work page 2021
[3]

Andrews, Y

B. Andrews, Y. Hu, and H. Li, Harmonic mean curvature flow and geometric inequalities , Adv. Math. 375 (2020), 107393

work page 2020
[4]

Blaschke, ¨Uber affine Geometrie I: Isoperimetrische Eigenschaften von Ellipse and Ellipsoid , Ber

W. Blaschke, ¨Uber affine Geometrie I: Isoperimetrische Eigenschaften von Ellipse and Ellipsoid , Ber. Verh. S¨ achs. Akad. Wiss. Leipzig. Math.-Phys. KI.68 (1916), 217–239

work page 1916
[5]

, Vorlesungen ¨ uber Differentialgeometrie II: Affine Differentialgeometrie(1923)

work page 1923
[6]

Bourgain and V.D

J. Bourgain and V.D. Milman, New volume ratio properties for convex symmetric bodies in Rb, Invent. Math. 88 (1987), 319–340

work page 1987
[7]

Brendle, Constant mean curvature surfaces in warped product manifolds , Publ

S. Brendle, Constant mean curvature surfaces in warped product manifolds , Publ. Math. de l’IH ´ES 117 (2013), 247–269. AFFINE ISOPERIMETRIC TYPE INEQUALITIES FOR STATIC CONVEX DOMAINS 17

work page 2013
[8]

Brendle, P.K

S. Brendle, P.K. Hung, and M.T. Wang, A Minkowski-type inequality for hypersurfaces in the Anti- deSitter-Schwarzschild manifold, Comm. Pure Appl. Math. 69 (2016), no. 1, 124–144

work page 2016

Show all 40 references

[9]

Brendle and M.T

S. Brendle and M.T. Wang, A Gibbons-Penrose inequality for surfaces in Schwarzschild spacetime , Commun. Math. Phys. 330 (2014), 33–43

2014
[10]

Deicke, ¨Uber die Finsler-R¨ aume mitAi = 0, Arch

A. Deicke, ¨Uber die Finsler-R¨ aume mitAi = 0, Arch. Math. 4 (1953), 45–51

1953
[11]

de Lima and F

L.L. de Lima and F. Girao, An Alexandrov-Fenchel-Type Inequality in Hyperbolic Space with an Ap- plication to a Penrose Inequality , Annales Henri Poincare 17 (2016), 979–1002

2016
[12]

F. Gao, D. Hug, and R. Schneider, Intrinsic volumes and polar sets in spherical space , Math. Notae. (2003), 159–176

2003
[13]

Y. Ge, G. Wang, and J. Wu, Hyperbolic Alexandrov-Fenchel quermassintegral inequalities II , J. Dif- ferential Geom. 98 (2014), no. 2, 237–260

2014
[14]

Z, 281 (2015), 257–297

, The GBC mass for asymptotically hyperbolic manifolds , Math. Z, 281 (2015), 257–297

2015
[15]

Gibbons, Collapsing shells and the isoperimetric inequality for black holes , Class

G.W. Gibbons, Collapsing shells and the isoperimetric inequality for black holes , Class. Quantum Grav. 14 (1997), 2905–2915

1997
[16]

Hu and H

Y. Hu and H. Li, Geometric inequalities for static convex domains in hyperbolic space , Trans. Amer. Math. Soc. 375 (2022), no. 8, 5587–5615

2022
[17]

, Geometric inequalities for hypersurfaces with nonnegative sectional curvature in hyperbolic space, Calc. Var. Partial Differential Equations 58 (2019), no. 2, 55

2019
[18]

, Blaschke-Santal´ o type inequalities and quermassintegral inequalities in hyperbolic space, Adv. Math. 413 (2023), 108826

2023
[19]

Y. Hu, H. Li, and Y. Wei, Locally constrained curvature flows and geometric inequalities in hyperbolic space, Math. Ann. 382 (2022), no. 3-4, 1425–1474

2022
[20]

Y. Hu, Y. Wei, and T. Zhou, A Heintze-Karcher type inequality in hyperbolic space , J. Geom. Anal. 34 (2024), no. 4, Paper No. 113, 17

2024
[21]

Hug, Curvature relations and affine surface area for a general convex body and its polar , Results Math

D. Hug, Curvature relations and affine surface area for a general convex body and its polar , Results Math. 29 (1996), no. 3-4, 233–248

1996
[22]

Li and Y

H. Li and Y. Wan, The Christoffel problem in the hyperbolic hyperplane , Adv. in Appl. Math. 150 (2023), Paper No. 102557

2023
[23]

H. Li, Y. Wei, and C.-W. Xiong, A geometric inequality on hypersurface in hyperbolic space , Adv. Math. 253 (2014), 152–162

2014
[24]

Li and B

H. Li and B. Xu, A class of weighted isoperimetric inequalities in hyperbolic space , Proc. Amer. Math. Soc. 151 (2023), no. 5, 2155–2168

2023
[25]

, Hyperbolic p-sum and horospherical p-Brunn-Minkowski theory in hyperbolic space (2022), available at arXiv:2211.06875

2022 arXiv
[26]

Li and C

J. Li and C. Xia, An integral formula and its applications on sub-static manifolds , J. Differential Geom. 113 (2019), 493–518

2019
[27]

Lutwak, The Brunn-Minkowski-Firey theory

E. Lutwak, The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas , Adv. Math. 118 (1996), no. 2, 244–294

1996
[28]

Mars and A

M. Mars and A. Soria, Geometry of Normal Graphs in Euclidean Space and Applications to the Penrose Inequality in Minkowski , Ann. Henri Poincar´ e15 (2014), 1903–1918

2014
[29]

Petty, Affine isoperimetric problems, Ann

C.M. Petty, Affine isoperimetric problems, Ann. N.Y. Acad. Sci. 440 (1985), 113–127

1985
[30]

Qiu and C

G. Qiu and C. Xia, A generalization of Reilly’s formula and its applications to a new Heintze–Karcher type inequality, Int. Math. Res. Not. 2015(17) (2015), 7608–7619

2015
[31]

Reilly, Geometric applications of the solvability of Neumann problems on a Riemannnian mani- fold, Arch

R.C. Reilly, Geometric applications of the solvability of Neumann problems on a Riemannnian mani- fold, Arch. Ration. Mech. Anal. 75 (1980), no. 1, 23–29

1980
[32]

Santal´ o,Un invariante afin para los cuerpos convexos del espacio de n dimensiones, Portugaliae Math

L.A. Santal´ o,Un invariante afin para los cuerpos convexos del espacio de n dimensiones, Portugaliae Math. (1949), 155–161

1949
[33]

Scheuer and C

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

2019
[34]

Schneider, Convex Bodies: the Brunn-Minkowski Theory 151 (2014)

R. Schneider, Convex Bodies: the Brunn-Minkowski Theory 151 (2014). Second expanded ed., Ency- clopedia of Mathematics and its Applications

2014
[35]

Sch¨ utt and E

C. Sch¨ utt and E. Werner,Surface bodies and p-affine surface area , Adv. Math. 187 (2004), 98–145

2004
[36]

Wang and C

G. Wang and C. Xia, Isoperimetric type problems and Alexandrov-Fenchel type inequalities in the hyperbolic space 259 (2014), 532–546. 18 Y. HU, H. LI, Y. W AN, AND B. XU

2014
[37]

Wang and S.-T

M.-T. Wang and S.-T. Yau, Quasilocal mass in general relativity , Phys. Rev. Lett. 102 (2009), no. 2, 021101

2009
[38]

, Isometric embeddings into the Minkowski space and new quasi-local mass , Commun. Math. Phys. 288 (2009), no. 3, 919–942

2009
[39]

Werner and D

E. Werner and D. Ye, New Lp affine isoperimetric inequalities, Adv. Math. 218 (2008), 762–780

2008
[40]

Xia, A Minkowski type inequality in space forms , Calc

C. Xia, A Minkowski type inequality in space forms , Calc. Var. 55:96 (2016). School of Mathematical Sciences, Beihang University, Beijing 100191, P.R. China Email address: huyingxiang@buaa.edu.cn Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. C...

2016

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Agostiniani, M

V. Agostiniani, M. Fogagnolo, and L. Mazzieri, Minkowski Inequalities via Nonlinear Potential Theory, Archive for Rational Mechanics and Analysis 244 (2022), 51–85

work page 2022

[2] [2]

Andrews, X

B. Andrews, X. Chen, and Y. Wei, Volume preserving flow and Alexandrov-Fenchel type inequalities in hyperbolic space, J. Eur. Math. Soc.(JEMS) 23 (2021), no. 7, 2467–2509

work page 2021

[3] [3]

Andrews, Y

B. Andrews, Y. Hu, and H. Li, Harmonic mean curvature flow and geometric inequalities , Adv. Math. 375 (2020), 107393

work page 2020

[4] [4]

Blaschke, ¨Uber affine Geometrie I: Isoperimetrische Eigenschaften von Ellipse and Ellipsoid , Ber

W. Blaschke, ¨Uber affine Geometrie I: Isoperimetrische Eigenschaften von Ellipse and Ellipsoid , Ber. Verh. S¨ achs. Akad. Wiss. Leipzig. Math.-Phys. KI.68 (1916), 217–239

work page 1916

[5] [5]

, Vorlesungen ¨ uber Differentialgeometrie II: Affine Differentialgeometrie(1923)

work page 1923

[6] [6]

Bourgain and V.D

J. Bourgain and V.D. Milman, New volume ratio properties for convex symmetric bodies in Rb, Invent. Math. 88 (1987), 319–340

work page 1987

[7] [7]

Brendle, Constant mean curvature surfaces in warped product manifolds , Publ

S. Brendle, Constant mean curvature surfaces in warped product manifolds , Publ. Math. de l’IH ´ES 117 (2013), 247–269. AFFINE ISOPERIMETRIC TYPE INEQUALITIES FOR STATIC CONVEX DOMAINS 17

work page 2013

[8] [8]

Brendle, P.K

S. Brendle, P.K. Hung, and M.T. Wang, A Minkowski-type inequality for hypersurfaces in the Anti- deSitter-Schwarzschild manifold, Comm. Pure Appl. Math. 69 (2016), no. 1, 124–144

work page 2016