Constrained convex bodies with extremal affine surface areas

C. Sch\"utt; E. M. Werner; H. Huang; O. Giladi

REVIEW 3 major objections 4 minor 62 references

Constrained convex bodies with extremal affine surface areas

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Pith's one-line read For convex bodies, extremal affine surface area tracks a power of volume, up to dimension-dependent constants.

desk verdict Good ideas, likely true results, but the main lower-bound proof has a genuine recentering gap that must be fixed. read the letter →

arxiv 1908.07897 v2 pith:KTEQXJHA submitted 2019-08-21 math.FA

O. Giladi , H. Huang , C. Schütt , E. M. Werner This is my paper · ORCID

classification math.FA MSC 52A2052A2352A40

keywords L_paffinesurfaceareaconvexbodiesextremalJohnellipsoidLownerthinshellestimateisotropicconstantisoperimetricinequality

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

Given a convex body $K\subseteq \mathbb{R}^n$, the paper defines $IS_p(K)$ as the largest $L_p$-affine surface area among all convex subsets of $K$, and $os_p(K)$ as the smallest among all convex bodies containing $K$. It proves that both quantities are, up to factors that depend only on the dimension, powers of the volume: $IS_p(K)$ behaves like $|K|^{(n-p)/(n+p)}$ for $0\le p\le n$, and $os_p(K)$ obeys the same proportionality for $-n

What carries the argument

The load-bearing object is the $L_p$-affine surface area $$as_p(K)=\int_{\partial K} \kappa(x)^{\frac{p}{n+p}}\langle x,N(x)\$rangle^{{\frac{n(p-1)}}${n+p}}\,d\mu(x),$$ an affine-invariant boundary integral whose $p=1$ case is the classical affine surface area. The proof combines the $L_p$-affine isoperimetric inequality, which supplies the universal upper bound and the ellipsoid equality case, with a thin-shell concentration estimate for isotropic bodies: at least half the volume of an isotropic $K$ lies in a shell of width $O(n^{1/3})$ around radius $L_K\sqrt{n}$. Intersecting $K$ with the ball of that radius gives an inscribed body whose spherical boundary has known curvature, which yields the lower bound. For the outer quantities the analogous object is the Lowner ellipsoid of $K$, the minimal-volume ellipsoid containing $K$, whose volume controls the constants in Theorem 3.6.

What would settle it

Compute $IS_1(K)/|K|^{(n-1)/(n+1)}$ for a sequence of increasingly thin rectangular boxes in $\mathbb{R}^n$ with $|K|=1$; the theorem predicts this ratio stays above an explicit positive constant depending only on $n$ and $L_K$, so a ratio falling below that constant for any $n$ would refute the claim. A more direct check of the proof's premise is to test the shell estimate on an explicit isotropic body: if less than half the volume of a unit-volume isotropic cube lies in $\{x: |\|x\|-L_K\sqrt{n}|<cL_K n^{1/3}\}$ for the paper's constant $c$, the lower-bound argument as written fails.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is Theorem 3.5: for all $n$, all $0\le p\le n$, and every convex body $K$, $$\frac{1}{$n^{{5/6}}$}\left(\frac{C}{L_K}\right)^{\frac{2np}{n+p}}\frac{IS_p(B_2^n)}{|B_2^n|^{(n-p)/(n+p)}} \le \frac{IS_p(K)}{|K|^{(n-p)/(n+p)}} \le \frac{IS_p(B_2^n)}{|B_2^n|^{(n-p)/(n+p)}},$$ where $L_K$ is the isotropic constant of $K$, and the upper bound is attained, for $p\neq 0,n$, exactly when $K$ is a centered ellipsoid. Because the right-hand quotient is a known dimensional constant, $IS_p(K)$ is determined by $|K|$ up to an $n$-dependent factor. The companion Theorem 3.6 proves an outer version for $os_p(K)$ when $-n<p\le 0$, with the normalized quantity lying between the ball value and $n^{n(n-p)/(n+p)}$ times it, and centered ellipsoids again the unique extremal bodies; for centrally symmetric bodies the power improves to $n^{n(n-p)/(2(n+p))}$.

Load-bearing premise

The lower bound rests on the thin-shell concentration estimate: after affinely normalizing $K$ so its volume is one and its inertia is isotropic, at least half the volume lies in a spherical shell of width proportional to $n^{1/3}$ around radius $L_K\sqrt{n}$; if this concentration fails with the stated constants, the inscribed ball-intersection constructed in the proof would carry too little boundary measure.

Editorial extensions

If this is right

For fixed $p\in[0,n]$, the shape of $K$ affects $IS_p(K)$ only through volume and the isotropic constant; bodies of equal volume have inner extremal affine surface areas within a dimension-dependent factor.
The sharp upper bound makes the Euclidean ball optimal among bodies of fixed volume for the normalized quantity, and the only optimizers are centered ellipsoids.
At the endpoints the statement is exact: $IS_0(K)=n|K|$ and $IS_n(K)=n|B_2^n|$; the new content is the whole open interval.
For $-n<p\le 0$, the smallest $L_p$-affine surface area among bodies containing $K$ cannot fall below a fixed power of $|K|$, and the Lowner ellipsoid provides the near-optimal outer body.
The monotonicity identities in Proposition 3.4 show the normalized quantities interpolate monotonically in $p$, so the volume-power law is part of a consistent $L_p$ family.

Reading between the lines

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

A direct test of the lower bound would be to compute $IS_p$ for explicit polytopes such as cubes: the cube has zero classical affine surface area, but its inner extremal value is at least that of the inscribed ball, so tracking the gap as $n$ grows would probe how sharp the $L_K$-dependent constant is.
Because the lower bound uses only the thin-shell half-volume shell, any improvement in the concentration constants would tighten the proportionality, suggesting that the sharp high-dimensional constant is governed by how close $K$ is to Gaussian volume concentration.
The equality characterization invites a stability version: bodies far from ellipsoids should have normalized $IS_p$ strictly below the ball value by an amount controlled by a measure of asymmetry; the paper does not quantify this, but its two-sided bounds are the natural starting point.

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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

Here's my read. The paper is worth engaging, but not in its current form. The central results are plausible and likely correct, but the main lower-bound proof has a gap that is not a typo.

What's genuinely new: the extremal inner and outer surface areas IS_p, os_p, OSp are natural objects, and this is the first higher-dimensional treatment. The p-range analysis is clean and useful, and the proportionality of these quantities to a power of volume is a real structural insight, not a rehash of known results. The upper bounds are fine, and the use of the thin shell estimate and Löwner ellipsoid is reasonable.

Now the soft spots. The sign error in (2.1) is explicit: the support exponent should be n(1-p)/(n+p), not n(p-1)/(n+p). With the printed formula the stated homogeneity (2.4) fails, so this is an internal inconsistency, not a matter of convention. The lower-bound computation in Theorem 3.5 also has an algebraic slip; the product of the curvature and support factors does not produce the exponent they write, and the later step uses a different relationship between surface area and volume that accidentally yields the advertised exponent. Both are fixable.

The serious issue is the recentering. The paper defines as_p only for bodies whose centroid is at the origin, and then evaluates it on the intersection K∩R B_2^n without recentering. After translating to the centroid, the support factor on the spherical cap is R−⟨g,θ⟩, not R. No lower bound for this factor over the relevant set of directions is supplied. The centroid shift g is not small relative to R in general; the isotropic condition only gives a bound that can be much larger than R for a small-volume cap. This is a genuine gap in the proof of Theorem 3.5, and the same problem appears in Theorem 3.6. The stress-test note lands.

I don't see a counterexample, and I'd bet the theorems are true. But the proof as printed doesn't close the case. A careful referee should ask for a revised proof, not reject the idea.

For peer review: yes, send it out. It deserves a serious referee. Would I cite it? Not until the proof is fixed. Reading group? Maybe, as a case study in why centroid assumptions matter.

Referee Report

3 major / 4 minor

Summary. The paper introduces extremal versions of L_p-affine surface area for a convex body K: IS_p(K), the supremum of as_p over convex subsets of K, and os_p(K), the infimum over convex bodies containing K, with analogous quantities OSp and isp. It identifies the meaningful p-ranges, proves upper semicontinuity/compactness and affine-isoperimetric inequalities for these functionals, and establishes, in the main Theorems 3.5 and 3.6, two-sided bounds showing that these extremal quantities are proportional to a power of |K| up to constants depending only on n. The proof of the lower bound in Theorem 3.5 uses the thin shell estimate of Guédon and Milman applied to K in isotropic position, constructing a subset K ∩ R B_2^n whose spherical boundary contributes to as_p. The paper also proves that these quantities are not quermassintegrals or valuations, complementing Bárány's two-dimensional result for p=1.

Significance. If the main results are correct, the paper opens a natural extremal analogue of John's theorem and the Löwner ellipsoid in the affine-surface-area setting, and it is surprising that both the inner and outer extremal quantities are proportional to a power of volume. The p-range classification, the continuity results, and the relation to quermassintegrals are useful and clearly presented. The paper relies on standard external tools (thin shell estimates, L_p-affine isoperimetric inequalities, John and Löwner ellipsoids) rather than fitted parameters, and the main theorems are stated as explicit inequalities with equality cases. This is a worthwhile contribution to affine convex geometry, provided the proof gaps identified below are repaired.

major comments (3)

[Section 2, Eq. (2.1)] The exponent of the support factor in the definition of L_p-affine surface area has the wrong sign. As printed, as_0(K) = ∫_{∂K} ⟨x,N(x)⟩^{-1} dμ, but the identity as_0(K)=n|K| used in Section 3.1 requires ∫_{∂K} ⟨x,N(x)⟩ dμ = n|K|, not the reciprocal integral. Equivalently, for T=λI, Eq. (2.4) gives as_0(λB)=λ^n as_0(B), whereas Eq. (2.1) gives λ^{n-2} as_0(B). Since the p=0 and p=n cases and all later integral computations depend on (2.1), this is a load-bearing error rather than a typographical one.
[Section 4, proof of Theorem 3.5] The displayed computation after 'as_p(K ∩ R B_2^n) ≥ ' contains an algebraic slip: the product R^{-(n-1)p/(n+p)} R^{n(p-1)/(n+p)} has exponent [-(n-1)p+n(p-1)]/(n+p), not [(n-1)p+n(p-1)]/(n+p). The subsequent exponent 2np/(n+p)-1 is therefore not the exponent of the integrand. For p=0, the integrand is R^{-1}, while the claimed expression is R. Because the final lower bound's powers of R and L_K are read off from this line, the computation must be redone after the sign in (2.1) is fixed.
[Section 4, proof of Theorem 3.5] The lower bound evaluates as_p(C) for C=K ∩ R B_2^n by integrating over the spherical part of ∂C with support factor R, but the definition (2.1) is only valid after translating C so that its center of gravity is at the origin. The set C need not be centered: after replacing C by C−g(C), the support factor on the corresponding spherical cap becomes R−⟨g(C),θ⟩, not R. The isotropic position of K gives only a bound of the form |g(C)| ≤ C n^{11/12} L_K (since |C| is bounded below by n^{-5/6}), which is not small compared with R≈√n L_K; no lower bound for R−⟨g(C),θ⟩ on a set of directions of positive measure is supplied. For p away from the endpoints, a relative deficit in this factor can substantially change the integral. The same issue affects the lower-bound computation in the proof of Theorem 3.6, where the constructed body conv{R²K, R B_2^n} is not shown to be centered. This is a gap in the central lower-bound derivation.

minor comments (4)

[Theorem 3.6] There are two parts labelled '(i)'; the second should be labelled '(ii)'.
[Lemma 3.2] In parts (ii) and (iii) of the proof, the approximating sequence is described as satisfying C_k ⊂ K; for the outer supremum OSp and outer infimum osp, the bodies must contain K, so the inclusion should be C_k ⊃ K.
[Section 2] The two families of convex sets are both denoted by KK; distinct symbols, for instance ℒ_K and ℒ^K, would avoid ambiguity.
[Proposition 3.7] The notation IS^β_1 and OS^β_{n/2} is introduced without a definition; the reader has to infer that these are powers of the corresponding functionals.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the main estimates are driven by external thin-shell and Lp-isoperimetric inequalities, not by the paper's own definitions.

full rationale

The paper's central result, Theorem 3.5, is not circular. The upper bound for IS_p(K) follows by applying the external Lp-affine isoperimetric inequality to each admissible subset K' of K, and the lower bound is obtained by explicitly constructing a subset of the form K ∩ R B_2^n and estimating its affine surface area using the Guédon–Milman thin-shell estimate. No parameter is fitted to IS_p(K), and the conclusion does not reduce to its input by construction. The cited prior results include papers by the authors, notably Werner–Ye for the Lp-affine isoperimetric inequality, but that inequality is an independent published theorem that does not assume or encode the newly defined quantities IS_p or os_p; hence these self-citations are not load-bearing in a circular sense. The potential issues raised by the skeptic, concerning whether K ∩ R B_2^n is centered as required by definition (2.1), and the apparent exponent algebra in the lower-bound estimate, are correctness or rigor concerns about the proof, not instances of the conclusion being equivalent to the assumptions. No circular step was found, so the appropriate score is the low end of the no-significant-circularity range.

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

All bounds are derived from external theorems; no parameters are fitted. The isotropic constant L_K is a property of K, not a fitted value. The new functionals IS_p and os_p are defined from existing objects and introduce no new physical or mathematical entities.

assumptions (5)

domain assumption Lp-affine isoperimetric inequality: for p>=0, asp(K)/asp(B_2^n) <= (|K|/|B_2^n|)^{(n-p)/(n+p)}, equality iff ellipsoid (and reverse for -n<p<=0).
Invoked in Proposition 3.4 and Theorem 3.5 upper bounds; cited from Lutwak [38] and Werner-Ye [60].
domain assumption Guedon-Milman thin shell estimate (4.3): for an isotropic convex body with |K|=1, at least 1/2 of its volume lies in the shell {x: |norm(x)-L_K sqrt(n)| < c L_K n^{1/3}}.
Load-bearing for the lower bound in Theorem 3.5; cited from [23].
domain assumption Bourgain-Milman volume product inequality |K||K^o| >= c^n |B_2^n|^2.
Used in proof of Theorem 3.6 to lower-bound |R^2 K|.
standard math John's and Lowner ellipsoid sandwiching: E in K in nE for the maximal volume ellipsoid, and K in L in nK for the Lowner ellipsoid (sqrt(n)K for symmetric K).
Used in Lemma 3.2 and Theorem 3.6.
standard math Blaschke selection theorem and upper/lower semicontinuity of asp in Hausdorff metric.
Used to show the extrema are attained in Lemma 3.2; continuity is Proposition 3.1.

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Cite this review

Pith. "Pith review of Constrained convex bodies with extremal affine surface areas." pith.science (2026). https://pith.science/paper/KTEQXJHA

@misc{pith2026190807897,
  author       = {Pith},
  title        = {Pith review of: Constrained convex bodies with extremal affine surface areas},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KTEQXJHA}},
  note         = {Machine review of arXiv:1908.07897}
}

read the original abstract

Given a convex body K in R^n and p in R, we introduce and study the extremal inner and outer affine surface areas IS_p(K) = sup_{K'\subseteq K} (as_p(K') ) and os_p(K)=inf_{K'\supseteq K} (as_p(K') ), where as_p(K') denotes the L_p-affine surface area of K', and the supremum is taken over all convex subsets of K and the infimum over all convex compact subsets containing K. The convex body that realizes IS_1(K) in dimension 2 was determined by Barany. He also showed that this body is the limit shape of lattice polytopes in K. In higher dimensions no results are known about the extremal bodies. We use a thin shell estimate of Guedon and Milman and the L\"owner ellipsoid to give asymptotic estimates on the size of IS_p(K) and os_p(K). Surprisingly, both quantities are proportional to a power of volume.

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