Some inequalities of isoperimetric type for the c-affine surface area

Arnon Chor; Shiri Artstein-Avidan

REVIEW 2 major objections 3 minor 11 references

Some inequalities of isoperimetric type for the c-affine surface area

T0 review · 2 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For ball-bodies, the c-affine surface area is maximized uniquely by the ball of radius n/(n+1), and the Santaló-type product is maximized uniquely by the half-ball.

desk verdict The main inequalities are real and the proofs are mostly clean, but the equality case of Theorem 7 is not justified, and that gap propagates to several equality characterizations. read the letter →

arxiv 2505.19172 v4 pith:KMIOGT64 submitted 2025-05-25 math.MG

Shiri Artstein-Avidan , Arnon Chor This is my paper · ORCID

classification math.MG MSC 52A2052A4052A38

keywords c-affinesurfaceareaball-bodiesSantaló-typeinequalityc-dualityprincipalradiiofcurvatureisoperimetric-typeinequalitiesfloatingbodiesconvexgeometry

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 sharp isoperimetric-type inequalities for the c-affine surface area $\Omega^c$, a functional defined on ball-bodies, the convex sets that are intersections of translates of the unit ball. The main results are that $\Omega^c(K)$ is maximized, uniquely, by the ball of radius $\frac{n}{n+1}$, and that the Santaló-type product $\Omega^c(K)\Omega^c(K^c)$ is maximized, uniquely, by the ball of radius $\frac{1}{2}$. These bounds matter because $\Omega^c$ arises from a c-floating-body construction and plays the role of an affine surface area in a class where the classical one is not naturally adapted. The proofs rest on a duality relation between the principal radii of curvature of a body and its c-dual, combined with Hölder's inequality and classical surface-area inequalities.

What carries the argument

The engine is the c-duality $K^c=\bigcap_{x\in K}(x+B_2^n)$, an involution on ball-bodies with $K-K^c=B_2^n$. Its analytic content is Theorem 6: at almost every direction $u$, the principal radii $r_i(u)$ of $K$ and $s_i(-u)$ of $K^c$ satisfy $r_i(u)+s_{n-i}(-u)=1$. This identity rewrites $\Omega^c(K^c)$ as the integral of the mirror product $r_i^{1/(n+1)}(1-r_i)^{n/(n+1)}$, so that Hölder's inequality with suitably chosen interpolating functions produces the surface-area comparison of Theorem 7. A second, simpler mechanism is the pointwise fact that $r\mapsto (1-r)^{1/(n+1)}r^{n/(n+1)}$ on $(0,1)$ is maximized at $r=\frac{n}{n+1}$, which already proves Theorem 3 in one line.

What would settle it

Use the standard equality condition for Hölder's inequality in the proof of Theorem 7: it gives $\prod_{i=1}^{n-1}\frac{r_i(u)}{1-r_i(u)}=\text{const}$ on $D_K$, not $\prod_{i=1}^{n-1}r_i(u)=\text{const}$; finding a non-ball body in $\mathcal{S}^n$ whose radii satisfy the first relation but not the second, or proving none exists, would settle whether the claimed equality characterizations are true.

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Extended reading notes

Core claim

On the class $\mathcal{S}^n$ of ball-bodies, the paper establishes two sharp results: Theorem 3, $\Omega^c(K)\le \Omega^c(\frac{n}{n+1}B_2^n)$ with equality only for $K=\frac{n}{n+1}B_2^n$, and Theorem 5, $\Omega^c(K)\Omega^c(K^c)\le \Omega^c(\frac{1}{2}B_2^n)^2$ with equality only for $K=\frac{1}{2}B_2^n$. The c-affine surface area is written in terms of principal radii as $\Omega^c(K)=\omega_n\int_{S^{n-1}}\prod_{i=1}^{n-1}(1-r_i(u))^{1/(n+1)}r_i(u)^{n/(n+1)}\,d\sigma(u)$. The central interpolation inequality (Theorem 7) is $\Omega^c(K)\le S(K)^{(n-1)/n}\Omega^c(K^c)^{1/n}$, with equality only for balls; iterating it and then using a mixed-volume Brunn-Minkowski inequality yields the Santaló-type product bound.

Load-bearing premise

The load-bearing step is the equality case of the main interpolation inequality: the paper asserts, without a full proof, that equality in Hölder's inequality forces the product of the principal radii to be constant on a set of full measure, and if that implication is false, the uniqueness statements in the equality cases do not follow.

Editorial extensions

If this is right

In every dimension $n\ge 2$, the ball of radius $\frac{n}{n+1}$ is the unique maximizer of $\Omega^c$ among all non-degenerate ball-bodies, with no volume normalization needed in this class.
For every non-degenerate ball-body, $\Omega^c(K)\Omega^c(K^c)\le \Omega^c(\frac{1}{2}B_2^n)^2$, a Santaló-type statement for the c-dual pair.
Corollary 8 bounds the same product by $S(K)S(K^c)$, and hence by $S(\frac{1}{2}B_2^n)^2$, connecting the new functional to ordinary surface area.
Lemma 9 together with Proposition 10 yields a second proof of Theorem 3 through $\Omega^c(K)\le S(\frac{n}{n+1}B_2^n)^{n/(n+1)}S(\frac{1}{n+1}B_2^n)^{1/(n+1)}$.

Reading between the lines

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

A natural next step, not taken in the paper, is a stability version: the gap in Theorem 7 between $\Omega^c(K)$ and $S(K)^{(n-1)/n}\Omega^c(K^c)^{1/n}$ should quantify how far a ball-body is from being a ball.
The same duality identity $r_i+s_{n-i}=1$ suggests a whole family of inequalities obtained by varying the two exponents in the interpolating functions; the paper's Remark 11 shows that the symmetric exponent choice fails for the product at $n\ge 4$, so other pairings would need a different argument.
Since $\Omega^c(K)\le \Omega(K)$, the ratio $\Omega^c/\Omega$ records how much the boundary curvature exceeds the unit-ball value; the two extremal radii $\frac{n}{n+1}$ and $\frac{1}{2}$ can be read as balancing curvature against c-dual complementarity, an interpretation the authors leave implicit.

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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 paper's main inequalities are correct and proved with standard tools, but the equality case of Theorem 7 is not justified: the stated Hölder condition is wrong and the claimed conclusion doesn't follow. That gap affects the equality statements in Corollary 8, Lemma 9 and Theorem 5, though the inequalities themselves stand.

The paper studies the c-affine surface area Ω^c for ball-bodies, a functional introduced by Schütt, Werner and Yalikun. The main results are natural extremal questions: Theorem 3 says Ω^c is maximized by the ball of radius n/(n+1) — the proof is a clean pointwise maximization of r^{n/(n+1)}(1-r)^{1/(n+1)}. Theorem 7 gives the comparison Ω^c(K) ≤ S(K)^{(n-1)/n} Ω^c(K^c)^{1/n}, and Theorem 5 derives the Santaló-type product bound with equality claimed for the 1/2-ball. The proofs are short and mostly transparent, and the appendix provides the needed support-function identity from previous work, so there is no circularity.

The soft spot is exactly the equality analysis in Theorem 7. The text says equality in Hölder implies φ/η and η are linearly dependent, which is not the condition: with p=n/(n-1), q=n, equality requires (φ/η)^{n/(n-1)} = λ η^n a.e. Working through the definitions, that gives ∏ r_i = c ∏ (1-r_i), not a constant product. The claimed inference to ∏ r_i constant is therefore unsupported, and the 'after a calculation' comment hides the gap. Since Corollary 8, Lemma 9, and the equality case of Theorem 5 all lean on this step, their sharpness claims are not fully proven. The inequality parts are unaffected, and Theorem 3's equality case has an independent pointwise argument, so the damage is confined.

This is a fixable defect. The stress-test note sketches a plausible repair using the measure identity that the ratio condition implies. As it stands, the paper needs a corrected equality argument or a more modest statement of the sharpness claims. I'd still send it to a referee: the main inequalities are real, the subject is new, and a short note like this with one repaired section is worth engaging.

Who is it for: people working on ball-bodies, affine surface area, or Santaló-type inequalities. A reading group could use it to discuss how Hölder equality conditions actually work.

Referee Report

2 major / 3 minor

Summary. The paper studies the c-affine surface area Ω^c introduced by Schütt–Werner–Yalikun on the class S^n of ball-bodies (intersections of translates of the unit ball). It gives two isoperimetric-type results: Theorem 3 asserts that among all ball-bodies, Ω^c(K) is uniquely maximized by the ball of radius n/(n+1); Theorem 5 asserts a Santaló-type inequality Ω^c(K)Ω^c(K^c) ≤ Ω^c(1/2 B_2^n)^2, with equality only for the ball of radius 1/2. The proofs are based on a representation of Ω^c in terms of principal radii (Eq. (3)), a support-function identity relating the principal radii of K and K^c (Theorem 6, proved in an appendix), and a Hölder inequality argument (Theorem 7) leading to Corollary 8, Lemma 9, and Proposition 10.

Significance. The main results are new and, if correct, provide sharp extremal statements for a recently introduced functional. The one-line proof of Theorem 3 is elegant and rigorous, and the Hölder derivation of Theorem 7 is a clean reduction to the c-dual support-function identity. The paper is self-contained: Theorem 6 is proved in the appendix and no parameters are fitted. The main weakness is the equality case in Theorem 7, which is stated with an incorrect Hölder equality condition and an unsupported 'calculation'; because the uniqueness claims of Theorems 3 (second proof), 5, Corollary 8, and Lemma 9 depend on this equality case, the sharpness statements are not currently established.

major comments (2)

[Theorem 7, equality case] The equality case of Theorem 7 is not justified. Hölder's inequality with p=n/(n-1) and q=n requires, for equality, (φ/η)^p = λ η^q a.e., i.e., (φ/η)^{n/(n-1)} = λ η^n, not 'φ/η and η are linearly dependent' as stated. Substituting φ(u)=ω_n∏(1-r_i)^{1/(n+1)}r_i^{n/(n+1)} and η^n(u)=ω_n∏ r_i^{1/(n+1)}(1-r_i)^{n/(n+1)}, the correct condition reduces to ∏ r_i(u) = c ∏(1-r_i)(u) on D_K, and does not yield ∏ r_i(u) ≡ const as claimed. The inference that the surface area measures of K and a ball coincide, and hence that K is a ball, is therefore unsupported. This is a load-bearing gap in the proof of the equality characterization in Theorem 7.
[Corollary 8, Lemma 9, Theorem 5, second proof of Theorem 3] The equality statements in these results all rely on the equality case of Theorem 7. In particular, Theorem 5's uniqueness claim 'equality if and only if K = 1/2 B_2^n' uses the equality case of Corollary 8, which in turn uses Theorem 7; Lemma 9 and the second proof of Theorem 3 also inherit this dependency. A repaired argument for the equality case of Theorem 7 would restore these, for example by showing that the correct condition ∏r_i = c∏(1-r_i) is equivalent to μ_K = c μ_{K^c}(-·), then using h_K+h_{K^c}(-·) = 1 and Minkowski's uniqueness theorem, but no such argument appears in the manuscript.

minor comments (3)

[Abstract] The phrase 'uniquely maximized be a Euclidean ball' should read 'uniquely maximized by a Euclidean ball'; the same typo recurs in the abstract.
[Proposition 10 proof] The name 'Aleandrov' should be 'Alexandrov'.
[Theorem 3 proof] The one-line proof of Theorem 3 would benefit from explicitly stating that equality in the pointwise bound forces each principal radius to equal n/(n+1) almost everywhere on D_K, which then yields the ball via Minkowski uniqueness.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained and the cited prior Theorem 6 is reproved in the appendix.

full rationale

The paper's central derivation is self-contained against the external definition of the c-affine surface area given by Schütt, Werner, and Yalikun. Definition 1 and the equivalent integral expression in equation (3) are obtained by a standard change of variables from boundary curvature to principal radii, not by assuming any of the paper's conclusions. Theorem 3 is a pointwise optimization of the integrand in (3), so its claim does not reduce to its input. The Santaló-type Theorem 5 is derived from Theorem 7 via Hölder's inequality together with Theorem 6, and although Theorem 6 is cited from the authors' earlier work [3], a full proof is supplied in the appendix using only the identity h_K(x)+h_{K^c}(-x)=|x| from the definition of the c-dual; hence that citation is not load-bearing in a circular way. No parameter is fitted and no claimed maximum or product inequality is used as an assumption in deriving itself. The paper's equality-case inference in Theorem 7, from equality in Hölder's inequality to constancy of the product of principal radii, is questionable as a matter of correctness, but that is a proof gap rather than a circular reduction of the claim to its own input. The main inequalities rest on the external definition of Ω^c and on standard inequalities such as Hölder, Brunn-Minkowski, and Alexandrov, not on a self-citation chain or on a renamed version of the target result.

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

No free parameters are fitted. The paper relies on standard convex geometry facts (Alexandrov differentiability, Minkowski uniqueness, Brunn-Minkowski for surface area) and on Theorem 6, for which a proof is included in the appendix. No new entities are postulated.

assumptions (3)

domain assumption For a ball-body K, principal radii r_i(u) exist and lie in (0,1) almost everywhere; the c-dual satisfies K - K^c = B_2^n.
Used throughout; standard for the class S^n, supported by [3] and [11].
standard math Alexandrov differentiability of support functions holds almost everywhere, and principal radii satisfy r_i = 1/κ_i at normal points.
From Schneider [10]; used to pass between curvature and support function formulations.
standard math Theorem 6 (r_i + s_{n-i} = 1) holds for K and K^c.
Proven in the appendix using the support function identity h_K(x)+h_{K^c}(-x)=|x|; relies on Alexandrov second-order expansion.

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Cite this review

Pith. "Pith review of Some inequalities of isoperimetric type for the c-affine surface area." pith.science (2026). https://pith.science/paper/KMIOGT64

@misc{pith2026250519172,
  author       = {Pith},
  title        = {Pith review of: Some inequalities of isoperimetric type for the c-affine surface area},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KMIOGT64}},
  note         = {Machine review of arXiv:2505.19172}
}

abstract

We study the c-affine surface area $\Omega^c$, recently introduced by Sch\"utt, Werner and Yalikun. We show that on the class of ball-bodies, $\Omega^c$ is maximized by a ball of radius $\frac{n}{n+1}$, and that a Santal\'o-type inequality holds: $\Omega^c(K) \Omega^c(K^c) \leq \Omega^c(\frac{1}{2} B_2^n)^2$. We also produce some more intricate inequalities involving the surface area.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 10 canonical work pages

[1]

A full classification of the isometries of the class of ball-bodies

Shiri Artstein-Avidan, Arnon Chor, and Dan I. Florentin. A full classification of the isometries of the class of ball-bodies. arXiv preprint 2503.02613 , 2025

work page Pith review arXiv 2025
[2]

Florentin

Shiri Artstein-Avidan, Arnon Chor, and Dan I. Florentin. Isometries of the class of ball- bodies. Colloquium Mathematicum , 178(2), 2025

work page 2025
[3]

Florentin

Shiri Artstein-Avidan and Dan I. Florentin. An in-depth study of ball-bodies. arXiv preprint 2505.09200, 2025

arXiv 2025
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Shiri Artstein-Avidan, Apostolos Giannopoulos, and Vitali D. Milman. Asymptotic Geo- metric Analysis. Part I , volume 202 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, 2015

work page 2015
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K´ aroly Bezdek, Zsolt L´ angi, and M´ arton Nasz´ odi. Selected topics from the theory of intersections of balls. arXiv preprint 2411.10302 , 2024

work page arXiv 2024
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Bonnesen and W

T. Bonnesen and W. Fenchel. Theorie der konvexen K¨ orper. Springer-Verlag, Berlin-New York, 1974. Berichtigter Reprint

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Geometric inequalities , volume 285

Yurii D Burago and Viktor A Zalgaller. Geometric inequalities , volume 285. Springer Science & Business Media, 2013

work page 2013
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Gulbank D. Chakerian. Sets of constant relative width and constant relative brightness. Transactions of the American Mathematical Society , 129(1):26–37, 1967

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Carsten Sch¨ utt, Elisabeth M. Werner, and Diliya Yalikun. Floating bodies for ball-convex bodies. arXiv preprint 2504.15488 , 2025. 9

2025 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

A full classification of the isometries of the class of ball-bodies

Shiri Artstein-Avidan, Arnon Chor, and Dan I. Florentin. A full classification of the isometries of the class of ball-bodies. arXiv preprint 2503.02613 , 2025

work page Pith review arXiv 2025

[2] [2]

Florentin

Shiri Artstein-Avidan, Arnon Chor, and Dan I. Florentin. Isometries of the class of ball- bodies. Colloquium Mathematicum , 178(2), 2025

work page 2025

[3] [3]

Florentin

Shiri Artstein-Avidan and Dan I. Florentin. An in-depth study of ball-bodies. arXiv preprint 2505.09200, 2025

arXiv 2025

[4] [4]

Shiri Artstein-Avidan, Apostolos Giannopoulos, and Vitali D. Milman. Asymptotic Geo- metric Analysis. Part I , volume 202 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, 2015

work page 2015

[5] [5]

Selected topics from the theory of intersections of balls

K´ aroly Bezdek, Zsolt L´ angi, and M´ arton Nasz´ odi. Selected topics from the theory of intersections of balls. arXiv preprint 2411.10302 , 2024

work page arXiv 2024

[6] [6]

Bonnesen and W

T. Bonnesen and W. Fenchel. Theorie der konvexen K¨ orper. Springer-Verlag, Berlin-New York, 1974. Berichtigter Reprint

work page 1974

[7] [7]

Geometric inequalities , volume 285

Yurii D Burago and Viktor A Zalgaller. Geometric inequalities , volume 285. Springer Science & Business Media, 2013

work page 2013

[8] [8]

Chakerian

Gulbank D. Chakerian. Sets of constant relative width and constant relative brightness. Transactions of the American Mathematical Society , 129(1):26–37, 1967

work page 1967