The Logarithmic Minkowski Problem

Deane Yang; Erwin Lutwak; Gaoyong Zhang; K\'aroly J. B\"or\"oczky

The Logarithmic Minkowski Problem

T0 review · 0 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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

Pith's one-line read An even measure on the unit sphere is the cone-volume measure of an origin-symmetric convex body exactly when it satisfies the subspace concentration condition, settling the even logarithmic Minkowski problem.

desk verdict A clean, correct solution of the even logarithmic Minkowski problem; no load-bearing gaps found. read the letter →

arxiv 2502.05430 v1 pith:LWHQ6LPU submitted 2025-02-08 math.MG

Károly J. Böröczky , Erwin Lutwak , Deane Yang , Gaoyong Zhang This is my paper · ORCID

classification math.MG MSC 52A40

keywords cone-volumemeasurelogarithmicMinkowskiproblemLp-Minkowskisubspaceconcentrationconditionorigin-symmetricconvexbodyBrunn-Minkowskiinequalitysymmetrizationfinite-dimensionalBanachspace

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 establishes a complete existence criterion for the even logarithmic Minkowski problem. It shows that a nonzero even Borel measure on the unit sphere is the cone-volume measure of an origin-symmetric convex body in $\mathbb{R}^n$ precisely when it satisfies the subspace concentration condition: no linear subspace of dimension $k$ may carry more than $k/n$ of the total mass, and equality must be accompanied by a complementary subspace carrying the complementary share. Cone-volume measures are the $p=0$ case of the $L_p$-surface-area measures and are the only ones invariant under all volume-preserving linear maps, so the theorem pins down exactly which sphere data can arise from a finite-dimensional normed space. The paper proves existence by minimizing a logarithmic functional, with the equality case handled through a structural decomposition forced by Brunn\u2013Minkowski equality. Uniqueness is not treated.

What carries the argument

The load-bearing objects are the cone-volume measure, defined for a Borel set $\omega\subset S^{n-1}$ by $V_K(\omega)=\frac1n\int_{\nu_K^{-1}(\omega)} x\cdot\nu_K(x)\,dH^{n-1}(x)$, and the subspace concentration condition stated above. Existence is obtained by minimizing $\Phi_\mu(K)=\int\log h_K\,d\mu$ over origin-symmetric bodies of fixed volume: under strict concentration a minimizer exists via John's ellipsoid and a cross-polytope estimate, and a variational lemma converts the minimizer into a body whose cone-volume measure is $\mu$. The equality case is the delicate part: symmetrization with respect to a subspace $\xi$ shows equality in the concentration inequality can happen only if every fiber of $K$ parallel to $\xi^\perp$ has the same volume, and the Brunn\u2013Minkowski equality conditions then force $K$ to be the Minkowski sum $(\xi^\perp\cap K)+C$, so that the measure concentrates on $\xi$ and a complementary subspace $\xi'$. This decomposition drives the inductive sufficiency proof.

What would settle it

The central claim would be falsified by exhibiting a non-zero even Borel measure on $S^{n-1}$ that satisfies the subspace concentration condition yet is not the cone-volume measure of any origin-symmetric convex body; the equality-case decomposition $K=(\xi^\perp\cap K)+C$ is the step most readily probed, for instance by searching for a measure on $S^2$ with equality on a subspace through the origin but no complementary subspace carrying the complementary share, which Theorem 1.1 forbids from being a cone-volume measure.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: a non-zero finite even Borel measure $\mu$ on $S^{n-1}$ is the cone-volume measure of an origin-symmetric convex body in $\mathbb{R}^n$ if and only if $\mu$ satisfies the subspace concentration condition. The condition has two parts: for every subspace $\xi$ with $0<\dim\xi<n$, one has $\mu(\xi\cap S^{n-1})\le (\dim\xi/n)\mu(S^{n-1})$; and whenever equality holds, some complementary subspace $\xi'$ also attains equality, equivalently $\mu$ is concentrated on $S^{n-1}\cap(\xi\cup\xi')$. Necessity is shown by symmetrizing a body about $\xi$ and comparing cone-volumes, while sufficiency follows from solving a variational problem for the logarithmic functional $\Phi_\mu(K)=\int\log h_K\,d\mu$, with the equality case handled by induction and a Minkowski-sum decomposition of the body. On the paper's own terms, this settles the existence part of the even logarithmic Minkowski problem for arbitrary measure data.

Load-bearing premise

The proof's load-bearing step is the equality case of the Brunn\u2013Minkowski inequality: it assumes that when all cross-sections of the body parallel to $\xi^\perp$ have equal volume, the body must split into the Minkowski sum of one such cross-section and a convex set spanning a complementary subspace, and the necessity argument depends on this splitting.

Editorial extensions

If this is right

Every even Borel measure satisfying the subspace concentration condition is the cone-volume measure of some origin-symmetric convex body, so existence holds with no smoothness or strict positivity assumptions beyond the condition.
Every origin-symmetric convex body's cone-volume measure automatically obeys the subspace concentration condition, and equality on a subspace forces the measure to be concentrated on that subspace and a complementary one.
Because origin-symmetric convex bodies are unit balls of finite-dimensional Banach spaces, the theorem completely characterizes which measures on the sphere arise as cone-volume measures of unit balls of normed spaces.
The solution for measures does not follow from the smooth or function case by approximation, so the measure-data formulation is an essential part of the result.
The discrete case is included: the theorem gives necessary and sufficient conditions for prescribed cone-volumes of an origin-symmetric polytope.

Reading between the lines

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

[Editorial extension] The equality structure of the subspace concentration condition should classify extremal bodies: a body whose cone-volume measure saturates the inequality on a subspace $\xi$ must decompose as $(\xi^\perp\cap K)+C$, so realizing bodies come with a built-in product structure that could be probed computationally.
[Editorial extension] If a similar characterization is sought for non-even measures, the subspace concentration condition would likely need an additional hypothesis excluding measures concentrated on a closed hemisphere; testing that extension is a natural next step.
[Editorial extension] For discrete measures the condition is finitely checkable: one can test all subspaces spanned by atoms to verify existence, which may make the theorem useful as a constructive certificate in computational geometry.

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

The main theorem is the full characterization: a non-zero even Borel measure on the sphere is a cone-volume measure of an origin-symmetric convex body exactly when it satisfies the subspace concentration condition. That is the real payoff, and the proof delivers it. The variational framework, the symmetrization argument for necessity, and the induction for the equality case all check out. The paper is the published JAMS 2013 article, so the mathematics is already well vetted; I found no circularity and no fitted or invented entities.

What is genuinely new: the measure-valued existence theorem, not just the discrete planar case (Stancu) or the necessary inequality for polytopes (He–Leng–Li, Xiong). The subspace concentration condition being both necessary and sufficient is a sharp, checkable characterization. The proof is self-contained modulo standard convex-geometry tools, and the delicate equality case in Theorem 5.2 is handled correctly via Brunn–Minkowski equality conditions. The compactness argument in Theorem 6.3 using John's ellipsoid and the cross-polytope estimate is also sound. I specifically looked for a hidden assumption in the step where equality forces each fiber to be a translate of ξ⊥∩K and the translation locus to be convex; that step is justified.

Soft spots are minor. The paper does not address uniqueness, which is openly stated and is a separate problem. The continuity of the logarithmic functional under Hausdorff convergence is asserted rather than proved explicitly, but this is a standard detail and not a gap. Also, Theorem 7.3 states n≥1 while Theorem 6.3 says n≥2; the n=1 case is trivial, so this is purely cosmetic. The self-citations are background and carry none of the proof.

The paper is important, careful, and honest about its limitations. The citation pattern is appropriate. This deserves a serious referee—though in practice it already went through one at JAMS and has become a standard reference. I would bring it to a reading group and cite it in my own work. My recommendation: engage with it fully.

Referee Report

0 major / 4 minor

Summary. The paper solves the even logarithmic Minkowski problem: Theorem 1.1 states that a non-zero finite even Borel measure on S^{n-1} is the cone-volume measure of an origin-symmetric convex body in R^n if and only if it satisfies the subspace concentration condition. The sufficiency is proved variationally: one minimizes the logarithmic functional Φ_μ(K)=∫ log h_K dμ under a fixed volume constraint, proves existence of a minimizer by a compactness argument (Theorem 6.3) that uses John's ellipsoid, a cross-polytope estimate (Lemma 6.2), and Blaschke selection, and then shows (Lemma 4.1) that any minimizer has cone-volume measure μ. Equality cases in the subspace concentration inequality are handled by induction on dimension, using Lemmas 7.1 and 7.2 to glue together bodies in complementary subspaces. The necessity of the subspace concentration condition is proved in Theorem 5.2 by a subspace symmetrization argument and an analysis of equality in the Brunn-Minkowski inequality.

Significance. If the result is correct, Theorem 1.1 completely resolves the existence part of the even logarithmic Minkowski problem, a central open problem in convex geometry. The subspace concentration condition is a clean, intrinsic condition that is both necessary and sufficient, so the theorem provides a definitive answer to a question that had only partial solutions for polytopes and smooth bodies. The proof is self-contained modulo standard tools (Brunn-Minkowski, John's theorem, Blaschke selection, Aleksandrov's lemma) and is notable for treating measure data directly rather than by approximation from the smooth case, which the authors explicitly explain is not possible. The equality-case analysis that characterizes the subspace concentration condition (not just the inequality) is a substantial contribution in its own right. The paper also makes the connection to finite-dimensional Banach spaces explicit through the correspondence with origin-symmetric convex bodies.

minor comments (4)

[Theorem 5.2 (equality case)] The step concluding that each fiber (x+ξ⊥)∩K is a translate of ξ⊥∩K from constant fiber volumes invokes the equality conditions of the Brunn-Minkowski inequality without explicitly stating them; adding a reference or a one-sentence justification would improve the self-containedness of the necessity argument.
[Theorem 6.3] Continuity of Φ_μ on K_n^e under Hausdorff convergence is used implicitly when passing from Blaschke selection to the existence of a minimizer; this follows from uniform convergence of support functions and the fact that the limiting body contains the origin in its interior, but the paper does not state it explicitly.
[Theorem 7.3] The theorem is stated for n≥1, but the proof in the strict case invokes Theorem 6.3, which is stated only for n≥2; the n=1 case is trivial (an even measure on S^0 is the cone-volume measure of a centered interval), but the proof should either split off n=1 or extend Theorem 6.3.
[Theorem 6.3 (contradiction step)] The implication from unboundedness of Φ_μ(C'_l) to unboundedness of Φ_μ(Q_l) is not written out; it uses h_{C_l}≤h_{Q_l} and the identity Φ_μ(C_l)=Φ_μ(C'_l)+(1/n)log γ, and spelling this out would remove a small gap for the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is proved from standard external results plus its own variational argument.

full rationale

The derivation is self-contained against external benchmarks. Theorem 1.1 is not assumed as an input: its necessity is proved in Theorem 5.2 from symmetrization, Fubini's theorem, and the equality case of Brunn-Minkowski, and its sufficiency is proved in Theorem 7.3 from the variational Lemma 4.1, the compactness argument in Theorem 6.3 (using John's ellipsoid and Lemma 6.2), and a lower-dimensional induction glued by Lemma 7.2. Citations to the authors' own earlier work (e.g., [24] for the Aleksandrov variational lemma and [39] for the Lp-Minkowski framework) are background or standard results; no load-bearing claim is justified solely by a self-citation. Lemma 4.1 derives dµ = h_K dS_K/n directly from the first variation of the logarithmic functional, with no fitted parameter renamed as a prediction. Lemma 6.2 and Theorem 6.3 establish existence of a minimizer by proving unboundedness of the functional off compact sequences, not by assuming the desired measure identity. The equality-case argument in Theorem 5.2, the most delicate step, uses standard Brunn-Minkowski equality conditions to extract the complementary subspace; it does not import the conclusion of the theorem. No quantity is defined in terms of another in a way that makes the cone-volume identity tautological, and no prediction is fitted to data. The only noted discrepancy is the harmless n≥1 versus n≥2 statement in Theorem 7.3, which is not a circularity concern.

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

The proof rests on standard theorems of convex geometry and measure theory. No free parameters are fitted, no new entities are postulated, and the subspace concentration condition is stated directly as the theorem's criterion rather than introduced as an ad hoc device.

assumptions (5)

standard math Brunn-Minkowski inequality and its equality conditions
Used in Section 5 to ensure the symmetrization S_ξK is convex and to characterize equality in Theorem 5.2.
standard math Blaschke selection theorem
Used in Theorem 6.3 to extract a convergent subsequence from a bounded minimizing sequence in K_e^n.
standard math John's theorem on maximal volume ellipsoids
Used in Theorem 6.3 to associate an ellipsoid E_l with each volume-normalized body Q_l and to construct the controlling cross-polytope C_l.
standard math Aleksandrov's lemma and weak continuity of surface area measures
Used in Section 3 and Lemma 4.1 to compute the first variation of volume and to derive the Euler-Lagrange equation dμ = (1/n)h_K dS_K.
standard math Riesz representation theorem for Borel measures on the sphere
Used implicitly in Lemma 4.1 to pass from equality of integrals against all even continuous functions to equality of measures.

how reviews work

0 comments

Cite this review

Pith. "Pith review of The Logarithmic Minkowski Problem." pith.science (2026). https://pith.science/paper/LWHQ6LPU

@misc{pith2026250205430,
  author       = {Pith},
  title        = {Pith review of: The Logarithmic Minkowski Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LWHQ6LPU}},
  note         = {Machine review of arXiv:2502.05430}
}

read the original abstract

In analogy with the classical Minkowski problem, necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone-volume measure of the unit ball of a finite dimensional Banach space.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

On polynomial inequalities for cone-volumes of polytopes
math.MG 2025-06 conditional novelty 6.0 of 10

For any finite set of unit normals positively spanning R^n, the cone-volume set C_cv(U) is a path-connected semialgebraic set, and it equals the scaled matroid base polytope only for centrally symmetric parallelepipeds.

Reference graph

Works this paper leans on

64 extracted references · 62 canonical work pages · cited by 1 Pith paper

[1]

Alexandrov, Existence and uniqueness of a convex surface with a given int egral curvature, C

A.D. Alexandrov, Existence and uniqueness of a convex surface with a given int egral curvature, C. R. (Doklady) Acad. Sci. USSR (N.S.) 35 (1942), 131–134

work page 1942
[2]

Andrews, Gauss curvature ﬂow: the fate of the rolling stones , Invent

B. Andrews, Gauss curvature ﬂow: the fate of the rolling stones , Invent. Math. 138 (1999), 151–161

work page 1999
[3]

Andrews, Classiﬁcation of limiting shapes for isotropic curve ﬂows , J

B. Andrews, Classiﬁcation of limiting shapes for isotropic curve ﬂows , J. Amer. Math. Soc. 16 (2003), 443–459

work page 2003
[4]

Barthe, O

F. Barthe, O. Guedon, S. Mendelson, and A. Naor, A probabilistic approach to the geometry of the ln p -ball, Ann. of Probability 33 (2005), 480–513

work page 2005
[5]

Caﬀarelli, Interior W 2,p-estimates for solutions of the Monge-Amp` ere equation , Ann

L. Caﬀarelli, Interior W 2,p-estimates for solutions of the Monge-Amp` ere equation , Ann. of Math. 131 (1990), 135–150

work page 1990
[6]

Campi and P

S. Campi and P. Gronchi, The Lp-Busemann-Petty centroid inequality , Adv. Math. 167 (2002), 128–141

work page 2002
[7]

Chen, Lp Minkowski problem with not necessarily positive data , Adv

W. Chen, Lp Minkowski problem with not necessarily positive data , Adv. Math. 201 (2006), 77–89

work page 2006
[8]

Cheng and S.-T

S.-Y. Cheng and S.-T. Yau, On the regularity of the solution of the n-dimensional Minkowski problem, Comm. Pure Appl. Math. 29 (1976) 495–516

work page 1976

Show all 64 references

[9]

K. S. Chou, Deforming a hypersurface by its Gauss-Kronecker curvature , Comm. Pure Appl. Math. 38 (1985), 867–882

1985
[10]

Chou and X.-J

K.-S. Chou and X.-J. Wang, The Lp-Minkowski problem and the Minkowski problem in centroaﬃne geometry, Adv. Math. 205 (2006), 33–83

2006
[11]

Chow, Deforming convex hypersurfaces by the nth root of the Gaussian curvature , J

B. Chow, Deforming convex hypersurfaces by the nth root of the Gaussian curvature , J. Diﬀerential Geom. 22 (1985), 117–138

1985
[12]

Cianchi, E

A. Cianchi, E. Lutwak, D. Yang, and G. Zhang, Aﬃne Moser-Trudinger and Morrey-Sobolev in- equalities, Calc. Var. Partial Diﬀerential Equations 36 (2009), 419–436

2009
[13]

Colesanti, M

A. Colesanti, M. Fimiani, The Minkowski problem for torsional rigidity , Indiana. Univ. Math. J. 59 (2010), 1013–1039

2010
[14]

Firey, Shapes of worn stones , Mathematika 21 (1974), 1–11

W. Firey, Shapes of worn stones , Mathematika 21 (1974), 1–11. THE LOGARITHMIC MINKOWSKI PROBLEM 21

1974
[15]

Gage and R

M. Gage and R. Hamilton, The heat equation shrinking convex plane curves , J. Diﬀerential Geom. 23 (1986), 69–96

1986
[16]

Gardner, Geometric Tomography, 2nd edition, Encyclopedia of Mathematics and its Applica- tions, Cambridge University Press, Cambridge, 2006

R.J. Gardner, Geometric Tomography, 2nd edition, Encyclopedia of Mathematics and its Applica- tions, Cambridge University Press, Cambridge, 2006

2006
[17]

Gardner, The Brunn-Minkowski inequality , Bull

R.J. Gardner, The Brunn-Minkowski inequality , Bull. Amer. Math. Soc. 39 (2002), 355–405

2002
[18]

Gardner and G

R.J. Gardner and G. Zhang, Aﬃne inequalities and radial mean bodies , Amer. J. Math. 120 (1998), 505–528

1998
[19]

Gromov and V.D

M. Gromov and V.D. Milman, Generalization of the spherical isoperimetric inequality for uniformly convex Banach Spaces , Composito Math. 62 (1987), 263–282

1987
[20]

Gruber, Convex and discrete geometry , Grundlehren der Mathematischen Wissenschaften, 336

P.M. Gruber, Convex and discrete geometry , Grundlehren der Mathematischen Wissenschaften, 336. Springer, Berlin, 2007

2007
[21]

Guan and P

B. Guan and P. Guan, Convex hypersurfaces of prescribed curvatures , Ann. Math. 156 (2002), 655– 673

2002
[22]

Guan and C.-S

P. Guan and C.-S. Lin, On equation det(ui j+ δi ju) = upf on Sn, preprint No. 2000-7, NCTS in Tsing-Hua University, 2000

2000
[23]

Guan and X

P. Guan and X. Ma, The Christoﬀel-Minkowski problem I: Convexity of solution s of a Hessian equation, Invent. Math. 151 (2003), 553–577

2003
[24]

Haberl, E

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

2010
[25]

Haberl and F

C. Haberl and F. Schuster, General Lp aﬃne isoperimetric inequalities , J. Diﬀerential Geom. 83 (2009), 1–26

2009
[26]

Haberl and F

C. Haberl and F. Schuster, Asymmetric aﬃne Lp Sobolev inequalities, J. Funct. Anal. 257 (2009), 641–658

2009
[27]

Haberl, F

C. Haberl, F. Schuster, and J. Xiao, An asymmetric aﬃne P´ olya-Szeg˝ o principle, Math. Ann. 352 (2012), 517-542

2012
[28]

B. He, G. Leng, and K. Li, Projection problems for symmetric polytopes , Adv. Math. 207 (2006), 73–90

2006
[29]

C. Hu, X. Ma, and C. Shen, On the Christoﬀel-Minkowski problem of Firey’s p-sum, Calc. Var. Partial Diﬀerential Equations 21 (2004), 137–155

2004
[30]

D. Hug, E. Lutwak, D. Yang, and G. Zhang, On the Lp Minkowski problem for polytopes , Discrete Comput. Geom. 33 (2005), 699–715

2005
[31]

Jerison, A Minkowski problem for electrostatic capacity , Acta Math

D. Jerison, A Minkowski problem for electrostatic capacity , Acta Math. 176 (1996), 1–47

1996
[32]

M. Y. Jiang, Remarks on the 2-dimensional Lp-Minkowski problem, Adv. Nonlinear Stud. 10 (2010), 297–313

2010
[33]

John, Polar correspondence with respect to a convex region , Duke Math

F. John, Polar correspondence with respect to a convex region , Duke Math. J. 3 (1937), 355–369

1937
[34]

Klain, The Minkowski problem for polytopes , Adv

D. Klain, The Minkowski problem for polytopes , Adv. Math. 185 (2004), 270–288

2004
[35]

Lewy, On diﬀerential geometry in the large, I, (Minkowski problem ), Trans

H. Lewy, On diﬀerential geometry in the large, I, (Minkowski problem ), Trans. Amer. Math. Soc. 43 (1938), 258–270

1938
[36]

Ludwig, Ellipsoids and matrix-valued valuations , Duke Math

M. Ludwig, Ellipsoids and matrix-valued valuations , Duke Math. J. 119 (2003), 159–188

2003
[37]

Ludwig, General aﬃne surface areas , Adv

M. Ludwig, General aﬃne surface areas , Adv. Math. 224 (2010), 2346–2360

2010
[38]

Ludwig and M

M. Ludwig and M. Reitzner, A classiﬁcation of SL(n) invariant valuations , Ann. of Math. 172 (2010), 1219–1267

2010
[39]

Lutwak, The Brunn-Minkowski-Firey theory

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

1993
[40]

Lutwak and V

E. Lutwak and V. Oliker, On the regularity of solutions to a generalization of the Min kowski problem, J. Diﬀerential Geom. 41 (1995), 227–246

1995
[41]

Lutwak, D

E. Lutwak, D. Yang, and G. Zhang, Lp aﬃne isoperimetric inequalities , J. Diﬀerential Geom. 56 (2000), 111–132

2000
[42]

Lutwak, D

E. Lutwak, D. Yang, and G. Zhang, A new ellipsoid associated with convex bodies , Duke Math. J. 104 (2000), 375–390

2000
[43]

Lutwak, D

E. Lutwak, D. Yang, and G. Zhang, The Cramer-Rao inequality for star bodies , Duke Math. J. 112 (2002), 59–81. 22 K ´AROLY J. B ¨OR ¨OCZKY, ER WIN LUTW AK, DEANE YANG, AND GAOYONG ZHANG

2002
[44]

Lutwak, D

E. Lutwak, D. Yang, and G. Zhang, Sharp aﬃne Lp Sobolev inequalities, J. Diﬀerential Geom. 62 (2002), 17–38

2002
[45]

Lutwak, D

E. Lutwak, D. Yang, and G. Zhang, On the Lp-Minkowski problem , Trans. Amer. Math. Soc. 356 (2004), no. 11, 4359–4370

2004
[46]

Lutwak, D

E. Lutwak, D. Yang, and G. Zhang, Volume inequalities for subspaces of Lp, J. Diﬀerential Geom. 68 (2004), 159–184

2004
[47]

Lutwak, D

E. Lutwak, D. Yang, and G. Zhang, Optimal Sobolev norms and the Lp Minkowski problem , Int. Math. Res. Not. 2006, Art. ID 62987, 21 pp

2006
[48]

Lutwak and G

E. Lutwak and G. Zhang, Blaschke-Santal´ o inequalities, J. Diﬀerential Geom. 47 (1997), 1–16

1997
[49]

Minkowski, Allgemeine Lehrs¨ atze ¨ uber die convexen Polyeder, Nachr

H. Minkowski, Allgemeine Lehrs¨ atze ¨ uber die convexen Polyeder, Nachr. Ges. Wiss. G¨ ottingen, 198–
[50]

Naor, The surface measure and cone measure on the sphere of ln p , Trans

A. Naor, The surface measure and cone measure on the sphere of ln p , Trans. Amer. Math. Soc. 359 (2007), 1045–1079

2007
[51]

Naor and D

A. Naor and D. Romik, Projecting the surface measure of the sphere of ln p , Ann. Inst. H. Poincar´ e Probab. Statist. 39 (2003), 241–261

2003
[52]

Nirenberg, The Weyl and Minkowski problems in diﬀerential geometry in t he large, Comm

L. Nirenberg, The Weyl and Minkowski problems in diﬀerential geometry in t he large, Comm. Pure and Appl. Math. 6 (1953), 337–394

1953
[53]

Paouris, Concentration of mass on convex bodies , Geom

G. Paouris, Concentration of mass on convex bodies , Geom. Funct. Anal. 16 (2006), 1021–1049

2006
[54]

Paouris and E

G. Paouris and E. Werner, Relative entropy of cone measures and Lp centroid bodies, Proc. London Math. Soc. 104 (2012), 253–286

2012
[55]

Pogorelov, The Minkowski multidimensional problem , V.H

A.V. Pogorelov, The Minkowski multidimensional problem , V.H. Winston & Sons, Washington, D.C., 1978

1978
[56]

Schneider, Convex bodies: the Brunn-Minkowski theory , Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1993

R. Schneider, Convex bodies: the Brunn-Minkowski theory , Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1993

1993
[57]

Stancu, The discrete planar L0-Minkowski problem, Adv

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

2002
[58]

Stancu, On the number of solutions to the discrete two-dimensional L0-Minkowski problem, Adv

A. Stancu, On the number of solutions to the discrete two-dimensional L0-Minkowski problem, Adv. Math. 180 (2003), 290–323

2003
[59]

Stancu, Centro-aﬃne invariants for smooth convex bodies , Int

A. Stancu, Centro-aﬃne invariants for smooth convex bodies , Int. Math. Res. Not. 2012, doi:10.1093/imrn/rnr110

2012 doi
[60]

A. C. Thompson, Minkowski geometry , Encyclopedia of Mathematics and its Applications, Cam- bridge University Press, Cambridge, 1996

1996
[61]

Umanskiy, On solvability of two-dimensional Lp-Minkowski problem , Adv

V. Umanskiy, On solvability of two-dimensional Lp-Minkowski problem , Adv. Math. 180 (2003), 176–186

2003
[62]

Xiong, Extremum problems for the cone volume functional for convex polytopes, Adv

G. Xiong, Extremum problems for the cone volume functional for convex polytopes, Adv. Math. 225 (2010), 3214–3228

2010
[63]

Zhang, The aﬃne Sobolev inequality , J

G. Zhang, The aﬃne Sobolev inequality , J. Diﬀerential Geom. 53 (1999), 183–202. Alfr´ ed R ´ enyi Institute of Mathematics Hungarian Academy of Science s Email address : carlos@renyi.hu Polytechnic Institute of New York University, Brooklyn, Ne w York Email address : elutwak@...

1999
[219]

II, Teubner, Leipzig, 1911, p p

Gesammelte Abhandlungen, Vol. II, Teubner, Leipzig, 1911, p p. 103–121

1911

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

Alexandrov, Existence and uniqueness of a convex surface with a given int egral curvature, C

A.D. Alexandrov, Existence and uniqueness of a convex surface with a given int egral curvature, C. R. (Doklady) Acad. Sci. USSR (N.S.) 35 (1942), 131–134

work page 1942

[2] [2]

Andrews, Gauss curvature ﬂow: the fate of the rolling stones , Invent

B. Andrews, Gauss curvature ﬂow: the fate of the rolling stones , Invent. Math. 138 (1999), 151–161

work page 1999

[3] [3]

Andrews, Classiﬁcation of limiting shapes for isotropic curve ﬂows , J

B. Andrews, Classiﬁcation of limiting shapes for isotropic curve ﬂows , J. Amer. Math. Soc. 16 (2003), 443–459

work page 2003

[4] [4]

Barthe, O

F. Barthe, O. Guedon, S. Mendelson, and A. Naor, A probabilistic approach to the geometry of the ln p -ball, Ann. of Probability 33 (2005), 480–513

work page 2005

[5] [5]

Caﬀarelli, Interior W 2,p-estimates for solutions of the Monge-Amp` ere equation , Ann

L. Caﬀarelli, Interior W 2,p-estimates for solutions of the Monge-Amp` ere equation , Ann. of Math. 131 (1990), 135–150

work page 1990

[6] [6]

Campi and P

S. Campi and P. Gronchi, The Lp-Busemann-Petty centroid inequality , Adv. Math. 167 (2002), 128–141

work page 2002

[7] [7]

Chen, Lp Minkowski problem with not necessarily positive data , Adv

W. Chen, Lp Minkowski problem with not necessarily positive data , Adv. Math. 201 (2006), 77–89

work page 2006

[8] [8]

Cheng and S.-T

S.-Y. Cheng and S.-T. Yau, On the regularity of the solution of the n-dimensional Minkowski problem, Comm. Pure Appl. Math. 29 (1976) 495–516

work page 1976