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On the $m$th order $p$-affine capacity

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Pith's one-line read This paper develops the mth order p-affine capacity $C_{p,Q}$, computes its exact value on the Euclidean ball, and proves a chain of affine isoperimetric and isocapacity inequalities linking volume, p-variational capacity, mth order…

desk verdict A solid extension of p-affine capacity to the mth-order setting; the main constant rides on an imported unpublished inequality, but the paper deserves referee time if that dependency is verified. read the letter →

arxiv 2505.12573 v1 pith:JGSMURDR submitted 2025-05-18 math.FA math.DGmath.MG

Xia Zhou , Deping Ye This is my paper · ORCID
classification math.FAmath.DGmath.MG MSC 52A4052A3853A1546E3046E3528A75
keywords mthorderp-affinecapacityLpaffineSobolevinequality(LpQ)-projectionbodyp-integralsurfaceareaisoperimetricp-variationalLipschitzstarbodies
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 builds a theory of the mth order p-affine capacity $C_{p,Q}(K)$, a higher-order affine-invariant analogue of the classical p-capacity, for $p\in[1,n)$ and for any convex body $Q$ in the space of $1\times m$ matrices. The capacity is defined as an infimum over test functions that are at least $1$ on $K$ of an affine energy that integrates support-function weights over all matrix directions. The paper proves that this quantity is finite, computes it exactly on the unit Euclidean ball in terms of the volume of a polar projection body, and establishes a four-term inequality chain comparing the capacity with volume, p-variational capacity, mth order p-integral affine surface area, and $L_p$ surface area. A reader would care because affine-invariant capacities give stronger, linearly invariant forms of Sobolev and isoperimetric inequalities and unify several existing p-affine capacities.

What carries the argument

The load-bearing object is the $(L_p,Q)$-projection body $\Pi_{p,Q}K$, whose support function is defined by $h_{\Pi_{p,Q}K}(x)^p=\int_{S^{n-1}} h_Q(v^T x)^p\,dS_{K,p}(v)$, and its polar volume enters the normalization constant $d_{n,p}(Q)=(n\omega_n)^{-1}(nm V_{nm}(\Pi_{p,Q}^*B_2^n))^{-p/(nm)}$. The mth order p-integral affine surface area $\Phi_{p,Q}(K)$ is the same polar volume, and the affine energy whose infimum defines $C_{p,Q}(K)$ is the matrix-direction integral $(\int_{S^{nm-1}}\|h_Q(\nabla_{\mathbf{u}}f)\|_p^{-nm}\,d\mathbf{u})^{-1/(nm)}$. Together these objects convert the classical ball computation for $p$-capacity into an affine-invariant statement, and the imported rearrangement inequality for the matrix-direction energy is what turns the volume lower bound into the exact ball value.

What would settle it

Take $n=2$, $m=2$ and $Q$ the unit square in $M_{1,2}(\mathbb{R})$, and compute $C_{p,Q}(B_2^n)$ numerically by discretizing the equivalent smooth-capacity formula over radial test functions; compare the result with $n\omega_n d_{n,p}(Q)\left(\frac{n-p}{p-1}\right)^{p-1}$. A discrepancy larger than the discretization error would falsify the exact ball value and, with it, the normalization of the isocapacity chain.

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Extended reading notes

Core claim

The central claim is that $C_{p,Q}$ is a well-defined finite functional with an exact normalization on the unit ball: for $Q\in\mathcal{K}_o^{1,m}$ and $1<p<n$, $C_{p,Q}(B_2^n)=n\omega_n d_{n,p}(Q)\left(\frac{n-p}{p-1}\right)^{p-1}$, with the corresponding $p=1$ value $n\omega_n d_{n,1}(Q)$. Moreover, for every Lipschitz star body $K$ the normalized quantities satisfy\[\left(\frac{V_n(K)}{V_n(B_2^n)}\right)^{1/n}\le\left(\frac{C_{p,Q}(K)}{C_{p,Q}(B_2^n)}\right)^{1/(n-p)}\le\left(\frac{\Phi_{p,Q}(K)}{\Phi_{p,Q}(B_2^n)}\right)^{1/(n-p)}\le\left(\frac{S_p(K)}{S_p(B_2^n)}\right)^{1/(n-p)}.\]Equality holds for ellipsoids in the capacity-volume comparison and for origin-symmetric ellipsoids in the capacity-affine-surface-area comparison. This extends the earlier p-affine capacity of Xiao and the asymmetric p-affine capacity of Hong and Ye, and its proof rests on an imported mth-order affine rearrangement inequality that supplies the lower bound forcing the ball constant.

Load-bearing premise

The lower-bound proof imports the mth-order affine energy inequality of [32], and the $p=1$ value imports the projection-body convergence result of [61]; if either imported result fails, the exact ball constant and the capacity below it collapse.

Editorial extensions

If this is right

  • The capacity-volume inequality gives a sharp affine isocapacity inequality with the ball as extremal, so all ellipsoids are equality cases.
  • The full chain shows that for every Lipschitz star body the normalized affine capacity is squeezed between the volume ratio and the $L_p$ surface area ratio, transferring any bound on $L_p$ surface area to the capacity.
  • When $m=1$ and $Q$ is chosen as the segment associated to the asymmetric weight $\varphi_\tau$, the results recover the known asymmetric p-affine capacity inequalities, making the new theory a common generalization.
  • Because $\Phi_{p,Q}$ transforms by $|\det\phi|^{(n-p)/n}$ under $\phi\in GL(n)$, the entire chain is covariant under linear maps, giving genuinely affine-invariant sharp constants.
  • The exact ball value fixes the normalization constants needed if the mth order affine energy is used in higher-order affine Sobolev inequalities.

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 Minkowski-type problem seeking a convex body $K$ with prescribed mth order p-affine capacity, since $C_{p,Q}$ is monotone, homogeneous, continuous from above, and enjoys affine invariance.
  • Equality in the capacity-affine-surface-area inequality is proved only for origin-symmetric ellipsoids; classifying equality for non-symmetric $K$ or non-symmetric $Q$ is a plausible open refinement.
  • One could test the sharpness of the chain numerically in low dimensions, for example $n=2$, $m=2$ with $Q$ the unit square, by computing $\Phi_{p,Q}$ and $S_p$ for cubes and comparing ratios to the stated bounds.
  • The $p=1$ limit suggests a direct total-variation-type definition of $C_{1,Q}$, which would extend the theory to BV functions and non-smooth sets.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. This paper develops a theory of the mth order p-affine capacity C_{p,Q} for p∈[1,n) and Q∈K_o^{1,m}, defined by taking the infimum of the mth order p-affine energy over Sobolev functions f≥1 on K. The authors establish several equivalent definitions (Theorem 3.3, Corollary 3.4), fundamental properties (monotonicity, homogeneity, finiteness, symmetry, concavity in Q, subadditivity, translation invariance, affine invariance, boundary behavior, upper semicontinuity and continuity from above), and compute the exact value of C_{p,Q}(B_2^n) in Theorem 4.4. They then prove a chain of inequalities comparing volume, the new capacity, the p-variational capacity, the mth order p-integral affine surface area Φ_{p,Q}, and the L_p surface area for Lipschitz star bodies (Theorems 4.1, 4.2, 4.4, 5.5, 5.6, Proposition 5.7). The main results rely on the mth order affine Pólya–Szegő principle of Langharst–Roysdon–Zhao ([32]) and, for the p=1 normalization, on a convergence result from the authors' preprint [61].

Significance. If the external inputs hold, the paper gives a substantial and natural extension of the p-affine capacity to the mth-order setting, with sharp affine-invariant isocapacity inequalities and an exact ball value. The internal computations—homogeneity, affine invariance via the spherical change of variables, and the coarea argument in Theorem 5.5—are consistent and carefully presented. The paper also extends the (L_p,Q)-projection body and Φ_{p,Q} to Lipschitz star bodies, which is useful. Its main limitation is that the precise constants in Theorems 4.1–4.4 and 5.5 are imported from an unpublished preprint [32]; the p=1 limit in Theorem 4.4 can be made self-contained. There is also a fundamental-property proof gap (subadditivity) that needs correction.

major comments (2)
  1. [Section 4, Eq. (4.4)–(4.9), Theorem 4.4] The lower bound (4.4) is quoted verbatim from [32, (3.19)] and the upper bound in Theorem 4.2 from [32, Theorem 1.2]; [32] is an arXiv preprint and neither inequality is proved here. Since (4.4), together with (4.8), determines the exact ball value in Theorem 4.4 and then propagates through Theorem 5.5 and the final chain, this is a load-bearing external input. The authors should either prove the needed mth-order affine Pólya–Szegő inequality (or at least verify its hypotheses in this setting) or cite a published version with the exact constants. In addition, the p=1 step (4.11) currently uses [61, Proposition 4.2], an unpublished same-author preprint; for K=B_2^n this limit can be obtained directly from (4.12) and continuity of L^p norms, so the external reference should be replaced by the short direct argument.
  2. [Proposition 3.5(iii)] The proof of subadditivity is not valid. From f1∈A(K1) and f2∈A(K2) one gets f1+f2∈A(K1∪K2), but this inclusion alone does not imply C_{p,Q}(K1∪K2)≤C_{p,Q}(K1)+C_{p,Q}(K2), because the functional E_{p,Q}(f)^p is not subadditive in f; the negative-exponent integral over S^{nm−1} does not satisfy such an inequality, even for disjointly supported f and g with anisotropic gradients (the m=1 case already shows the obstruction). The authors should either supply a correct proof (e.g., via a partition-of-unity argument if the statement is true) or remove the subadditivity claim.
minor comments (5)
  1. [Proposition 3.5(v)] The boundary-capacity identity is proved by gluing g=max{f,1} on K and g=f outside K. This function generally has a jump across ∂K and need not lie in W^{1,p}_0 unless the trace of f on ∂K equals 1 a.e. A correct argument is to take g=max{1,f} globally; then g≥1 on K and |∇g|≤|∇f| pointwise.
  2. [Theorem 4.4, proof of (4.11)] Once (4.12) is established, the convergence h_{Π_{p,Q}B_2^n}(u) → h_{Π_{1,Q}B_2^n}(u) follows directly from h_{Π_{p,Q}B_2^n}(u)^p = ∫_{S^{n−1}} h_Q(v^T u)^p dv and standard L^p continuity; citing [61, Proposition 4.2] is unnecessary and should be replaced.
  3. [Proposition 5.7] The displayed identity attributed to [32, (3.3)] is a standard consequence of averaging over O(n): for fixed u, the integral (5.15) is independent of v and equals its average over v∈S^{n−1}. This makes the proof more self-contained.
  4. [Notation] There are minor notational inconsistencies, e.g., K_o^{n,m} in the abstract versus K^{n,m}_{(o)} in Section 2, and the formula for C_{p,Q}(B_2^n) in the abstract is typeset with an extra parenthesis. These should be corrected.
  5. [References] References [32] and [61] are arXiv preprints; if either has been accepted for publication, the final versions should be cited. In particular, the authors should confirm the exact statement and hypotheses of [32, (3.19)] and [32, Theorem 1.2] in print.

Circularity Check

1 steps flagged · score 4.0 of 10

Only the p=1 endpoint rests on a same-author unpublished convergence result; the p>1 derivation is independent and externally benchmarked.

  1. self citation load bearing [Section 4, proof of Theorem 4.4, derivation of (4.11)]
    "Combining with (3.2), (4.12) and [61, Proposition 4.2], one gets hΠp,QK(•)→hΠ1,QK(•) uniformly on Snm−1 as p→1+."

    The p=1 row of Theorem 4.4 is pinned by the limit lim_{p→1+} d_{n,p}(Q)=d_{n,1}(Q). That limit is obtained by invoking [61, Proposition 4.2], a convergence result for Orlicz projection bodies stated in an unpublished preprint by the same two authors plus a third. The paper does not prove this proposition, so the p=1 ball value C1,Q(B2^n)=nωn d_{n,1}(Q) and the p=1 part of the main chain rest on a self-citation rather than on an argument given in this paper. The p>1 results are independent of this step, so the circularity is limited to the endpoint.

full rationale

The mth order p-affine capacity is defined from first principles via an infimum, and its basic properties are proved in the paper. No parameter is fitted to data and no quantity called a prediction reduces to an input. The main lower bound in Theorem 4.1 uses [32, (3.19)], and Theorem 4.2 uses [32, Theorem 1.2]; these are by Langharst, Roysdon and Zhao, not by the present authors, so they are external evidence rather than self-citation. The exact ball value for 1<p<n is obtained by matching an upper bound from [32, Theorem 1.2] with a lower bound from [32, (3.19)], neither of which is written by the authors, so this is not circular. The only same-author citation that actually carries a conclusion is [61, Proposition 4.2], an unpublished preprint by Zhou, Ye and Zhang, used to identify lim_{p→1+} d_{n,p}(Q)=d_{n,1}(Q) and thereby the p=1 ball value. This is a genuine self-citation that is load-bearing for the p=1 endpoint, but it does not infect the p>1 derivation; moreover the chain inequalities for p∈(1,n) are independently established. Hence the paper is largely self-contained against external benchmarks, with a mild but real self-citation at the p=1 limit.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The central claim rests on standard tools from convex geometry, geometric measure theory, and Sobolev space theory, plus several specific theorems imported from the literature: the mth order affine Polya-Szego inequality ([32]), the convergence of Orlicz projection bodies ([61]), and regularity lemmas for Lipschitz star bodies ([36]). No free parameters are fitted; the constant d_{n,p}(Q) is defined, not optimized. The only invented entity is the capacity C_{p,Q} itself, which has an independent handle because it specializes to known p-affine capacities when m=1.

assumptions (5)
  • domain assumption The mth order affine Polya-Szego inequality of Langharst, Roysdon and Zhao [32, Theorem 1.2 and (3.19)], giving the lower bound (4.5) for the mth order affine energy in terms of superlevel volume.
    Used to prove Theorem 4.1 and the lower bound in Theorem 4.4. The result is imported unchanged; the paper does not reprove it.
  • domain assumption Continuity of the Orlicz projection body operator as p tends to 1+, [61, Proposition 4.2] and the properties of h_Q^p in [61, P.10].
    Used to establish (4.11) and hence the p=1 value of the ball capacity. This is a same-author unpublished preprint.
  • domain assumption Properties of Lipschitz star bodies: finiteness of boundary measure, a.e. existence of radial derivative, the bounds a_K <= z dot nu_K(z) <= b_K [36, Lemma 2.1], and the Orlicz projection body lower bound (5.3) [36, Lemmas 6.1-6.2].
    Used to define L_p surface area for Lipschitz star bodies and to prove positivity and sublinearity of the extended projection body.
  • standard math Standard facts from geometric measure theory and Sobolev theory: Federer's coarea formula (2.11), Sard's theorem, density of C_c^infinity in W^{1,p}_0, the formula (3.5) for gradients of positive parts, and the p-variational capacity of the unit ball (4.2) from Maz'ya.
    Background assumptions used throughout; all are standard and cited to textbooks.
  • domain assumption The L_p (and Orlicz) projection body theory for convex bodies from [19] and [61], including the formula h_{Pi_{p,Q}K}(xxx)^p = integral h_Q(v^T xxx)^p dS_{K,p}(v) and the polarity relation (2.5).
    Used to define the normalizing constant d_{n,p}(Q) and the mth order p-integral affine surface area.
invented entities (2)
  • mth order p-affine capacity C_{p,Q}(K) independent evidence
    purpose: Central object of the paper; a variational functional over test functions measuring affine energy of superlevel sets; generalizes p-affine capacity and general p-affine capacity.
    It reduces to the known p-affine capacity of Xiao [55,56] and general p-affine capacity of Hong-Ye [23] when m=1 with Q the segment phi_tau, and it satisfies affine invariance and the sharp isocapacity inequalities proven in the paper, giving an independent handle outside the definition.
  • Extension of the mth order p-integral affine surface area Phi_{p,Q} (and the (L_p,Q)-projection body) to Lipschitz star bodies independent evidence
    purpose: Allows the isocapacity inequality chain to be stated for star bodies rather than only convex bodies.
    Definition 5.3 agrees with the convex-body definition from [18,19] on convex bodies and satisfies homogeneity and affine invariance (Proposition 5.4), giving a falsifiable handle.

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Pith. "Pith review of On the $m$th order $p$-affine capacity." pith.science (2026). https://pith.science/paper/JGSMURDR

@misc{pith2026250512573,
  author       = {Pith},
  title        = {Pith review of: On the $m$th order $p$-affine capacity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JGSMURDR}},
  note         = {Machine review of arXiv:2505.12573}
}
abstract

Let $M_{n, m}(\mathbb{R})$ denote the space of $n\times m$ real matrices, and $\mathcal{K}_o^{n,m}$ be the set of convex bodies in $M_{n, m}(\mathbb{R})$ containing the origin. We develop a theory for the $m$th order $p$-affine capacity $C_{p,Q}(\cdot)$ for $p\in[1,n)$ and $Q\in\mathcal{K}_{o}^{1,m}$. Several equivalent definitions for the $m$th order $p$-affine capacity will be provided, and some of its fundamental properties will be proved, including for example, translation invariance and affine invariance. We also establish several inequalities related to the $m$th order $p$-affine capacity, including those comparing to the $p$-variational capacity, the volume, the $m$th order $p$-integral affine surface area, as well as the $L_p$ surface area.

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.