{"id":"6e86c34c-7b31-4516-91c8-6ea328c728ef","arxiv_id":"2505.12573","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper defines the mth order p-affine capacity, proves its fundamental properties and exact unit ball value, and establishes affine isocapacity inequalities relating it to volume, p-capacity, mth order integral affine surface area, and L_p surface area.","lead":"This paper introduces a new geometric quantity, the mth order p-affine capacity, for convex bodies in the space of n-by-m matrices. It proves the quantity has the expected invariance and comparison properties, extending known affine isocapacity inequalities to a higher-order setting.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The main capacity inequalities inherit their key lower bound from an imported, not-reproven mth-order affine Pólya–Szegő inequality ([32,(3.19)]), and the p=1 ball value depends on an unpublished same-author convergence result ([61,Prop.4.2]); both need independent verification.","rationale":"The reader's weakest-assumption analysis correctly identifies reliance on [32] and [61]. I agree that this makes the paper conditional, but I would sharpen the concern: the genuinely load-bearing external step is [32,(3.19)] and its constant, since it drives Theorem 4.1, Theorem 4.4, and Theorem 5.5. The [61] dependency, by contrast, is removable for the unit ball by a short dominated-convergence argument, so a referee should ask for that proof rather than treat it as an equal load-bearing risk. I did not find a clear internal inconsistency in the new Section 3 material: the equivalence definitions, monotonicity, homogeneity, affine invariance, and semicontinuity arguments appear coherent, and the radial computations in Theorem 5.5 are dimensionally consistent. The main gap is therefore not the paper's own reasoning but the unproven imported inequality. Because the paper's headline theorems would collapse if [32,(3.19)] were false or misstated, the appropriate verdict is conditional, not accept or reject. This does not change the reader's verdict, hence UNCHANGED.","tokens_in":30409,"tokens_out":33669,"duration_ms":342230,"concrete_test":"Use the m=1 reduction to published results as an independent check of [32,(3.19)]. Choose Q so that h_Q(t)^p = \\phi_{\\tau}(t)^p, e.g., Q = [-((1-\\tau)/2)^{1/p}, ((1+\\tau)/2)^{1/p}], where C_{p,Q} reduces to the known general p-affine capacity C_{p,\\tau} of Hong–Ye (J. Geom. Anal. 28 (2018)). Recompute the right-hand side of Theorem 4.4, (4.6), for this Q and compare with the published exact value of C_{p,\\tau}(B_2^n) for 1<p<n. If the two values agree, the imported Pólya–Szegő constant is consistent with the established m=1 theory; if they differ, [32,(3.19)] or its application in Theorem 4.1 is incorrect and the central claim fails. A second, smaller check is to prove (4.11) directly by dominated convergence on S^{n-1}, removing the need for [61, Prop.4.2].","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim — that C_{p,Q} is finite with the exact ball value and the chain of inequalities in Sections 4–5 — rests on two imported results not proved in this paper. The lower bound in Theorem 4.1 is obtained verbatim from Langharst–Roysdon–Zhao [32, (3.19)], an arXiv preprint; Theorem 4.2 uses [32, Theorem 1.2]; and these feed directly into the exact ball value in Theorem 4.4 and into Theorem 5.5. If the constant, hypotheses, or normalization in [32, (3.19)] are wrong, Theorem 4.4 and the main chain fail. The second dependency, [61, Proposition 4.2], is an unpublished same-author preprint used only to identify the p=1 limit lim_{p→1+} d_{n,p}(Q) = d_{n,1}(Q). That step is much less load-bearing: for the unit ball, h_{\\Pi_{p,Q}B}(u) = (\\int_{S^{n-1}} h_Q(v^T u)^p dv)^{1/p}, so by L^p continuity on a finite measure space and uniform convergence, (4.11) follows directly without [61]. Thus the genuinely exposed point is the mth-order affine Pólya–Szegő inequality itself, which is nontrivial, unpublished, and used to fix the precise constants throughout Sections 4–5. The internal properties in Section 3 appear sound; the vulnerability is specifically the imported inequality and constant.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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].","tokens_in":30806,"tokens_out":29699,"duration_ms":294899,"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":[{"comment":"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.","section":"Section 4, Eq. (4.4)–(4.9), Theorem 4.4"},{"comment":"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.","section":"Proposition 3.5(iii)"}],"minor_comments":[{"comment":"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.","section":"Proposition 3.5(v)"},{"comment":"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.","section":"Theorem 4.4, proof of (4.11)"},{"comment":"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.","section":"Proposition 5.7"},{"comment":"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.","section":"Notation"},{"comment":"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.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is promising and the main chain of inequalities is plausible, but the exact-ball-value theorem rests on an unpublished preprint [32] by close collaborators, and the subadditivity proof is unsound as written. Before publication, the authors should either make the reliance on [32] transparent and verified, or include the necessary proof; the p=1 normalization can be made self-contained easily. I would not reject on the basis of the external dependence alone, but the subadditivity issue needs to be resolved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague —\n\nThis is a straightforward, competent extension paper. The authors define the mth order p-affine capacity C_{p,Q}, prove it has the standard regularity/invariance properties (homogeneity, affine invariance, subadditivity, upper semicontinuity, continuity from above), compute the ball value exactly, and string together the expected chain: volume ≤ capacity ≤ mth order p-integral affine surface area ≤ L_p surface area. The object is genuinely new; before this, the mth order theory stopped at the energy and projection body level. The proofs I checked internally are consistent: the sphere-change-of-variables in the affine invariance argument, the coarea step in Theorem 5.5, and the approximation arguments in Section 3 all look right. No fitted parameters, no circularity with their own results.\n\nThe soft spot is exactly where the stress-test puts it. The lower bound in Theorem 4.1 — and therefore the exact ball value in Theorem 4.4 and the whole chain of inequalities — is imported verbatim from Langharst–Roysdon–Zhao [32], an arXiv preprint. If that inequality's constant or hypotheses are off, this paper's headline theorem falls. The paper does not reprove it. A referee should check whether [32] has been accepted or should ask the authors to include a proof of the specific form they need. That is a real, load-bearing dependency, and it is the only one I would insist on.\n\nI disagree with the reader about [61]. The p=1 normalization uses [61, Prop. 4.2] for the continuity of projection bodies in p, but the specific use here — identifying lim_{p→1+} d_{n,p}(Q) — can be obtained directly by L^p continuity of h_Q on the sphere, as the stress-test note observes. So that dependency is minor. The authors could patch it in two lines, and a referee should suggest they do.\n\nThe paper is an extension, not a breakthrough. For someone working in affine Sobolev theory or convex geometry, it is a useful contribution and it will likely become a standard citation. I would send it to peer review, with one clear request: make the dependency on [32] explicit and either provide a proof of the imported inequality or cite a published version.","headline":"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.","tokens_in":31332,"tokens_out":2947,"would_cite":false,"duration_ms":27836,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A40","52A38","53A15","46E30","46E35","28A75"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["mth order p-affine capacity","p-affine capacity","Lp affine Sobolev inequality","(Lp,Q)-projection body","mth order p-integral affine surface area","affine isoperimetric inequality","p-variational capacity","Lipschitz star bodies"],"falsifier":"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.","tokens_in":30226,"feed_emoji":"📐","tokens_out":8367,"duration_ms":81762,"temperature":0.7,"pith_summary":"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.","feed_headline":"Exact ball values found for a higher-order affine capacity","feed_subtitle":"A chain of inequalities links volume, capacity, affine surface area, and Lp surface area for star bodies.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the mth-order affine rearrangement inequality used for the lower bound in Theorem 4.1 and the comparison in Theorem 4.2.","marker":"[32]"},{"why":"Provides the convergence of Orlicz projection bodies as $p\\to 1^+$ that pins down the $p=1$ ball-value constant $d_{n,1}(Q)$.","marker":"[61]"},{"why":"Defines the $(L_p,Q)$-projection body and establishes the higher-order $L_p$ affine Sobolev and isoperimetric inequalities that the new capacity inequalities extend.","marker":"[19]"},{"why":"Gives the p-affine capacity special case and the proof techniques for the ball-value computation and the inequalities in Sections 4 and 5.","marker":"[23]"},{"why":"Supplies the $L_p$ surface-area comparison for Lipschitz star bodies used to close the final inequality in the chain.","marker":"[37]"},{"why":"Provides the classical p-variational capacity of the Euclidean ball, the normalization against which $d_{n,p}(Q)$ is scaled.","marker":"[44]"},{"why":"Proves the Orlicz projection-body lemmas for star bodies used to show that $\\Pi_{p,Q}K$ is a convex body with finite positive support function.","marker":"[36]"}],"fun_headline_variants":["Exact ball norm and volume-capacity bounds for affine capacity","Higher-order affine capacity: exact ball constants and sharp links","m-th order p-affine capacity: exact ball values and inequalities","Volume-capacity-surface area chain for m-th order p-affine capacity","Sharp affine capacity inequalities: from volume to Lp surface area"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Exact ball norm and volume-capacity bounds for affine capacity","Higher-order affine capacity: exact ball constants and sharp links","m-th order p-affine capacity: exact ball values and inequalities","Volume-capacity-surface area chain for m-th order p-affine capacity","Sharp affine capacity inequalities: from volume to Lp surface area"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000792,"raw_usage":{"total_tokens":3517,"prompt_tokens":1002,"completion_tokens":2515,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":618,"completion_tokens_details":{"reasoning_tokens":2427}},"tokens_in":618,"tokens_out":2515,"duration_ms":16786,"temperature":1.0,"reasoning_tokens":2427,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:32:57.545479+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"On the $m$th-order Affine P\\'olya-Szeg\\\"o Principle","cited_arxiv_id":"2409.02232","evidence_quote":"Supplies the mth-order affine rearrangement inequality used for the lower bound in Theorem 4.1 and the comparison in Theorem 4.2."},{"cited_title":"The $m$th order Orlicz projection bodies","cited_arxiv_id":"2501.07565","evidence_quote":"Provides the convergence of Orlicz projection bodies as $p\\to 1^+$ that pins down the $p=1$ ball-value constant $d_{n,1}(Q)$."},{"cited_title":"Haddad, D","cited_arxiv_id":null,"evidence_quote":"Defines the $(L_p,Q)$-projection body and establishes the higher-order $L_p$ affine Sobolev and isoperimetric inequalities that the new capacity inequalities extend."},{"cited_title":"Hong and D","cited_arxiv_id":null,"evidence_quote":"Gives the p-affine capacity special case and the proof techniques for the ball-value computation and the inequalities in Sections 4 and 5."},{"cited_title":"Ludwig, J","cited_arxiv_id":null,"evidence_quote":"Supplies the $L_p$ surface-area comparison for Lipschitz star bodies used to close the final inequality in the chain."},{"cited_title":"Maz’ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations , 2nd edn","cited_arxiv_id":null,"evidence_quote":"Provides the classical p-variational capacity of the Euclidean ball, the normalization against which $d_{n,p}(Q)$ is scaled."},{"cited_title":"Lin and D","cited_arxiv_id":null,"evidence_quote":"Proves the Orlicz projection-body lemmas for star bodies used to show that $\\Pi_{p,Q}K$ is a convex body with finite positive support function."}],"review_version":1}