{"id":"a4172757-885d-478a-849f-27b3d329216e","arxiv_id":"1908.07897","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For each convex body and parameter p in the meaningful range, the largest or smallest p-affine surface area among bodies inside or outside it grows like a fixed power of its volume, up to dimension-dependent constants.","lead":"Researchers introduce two new ways to measure the size of a convex shape's boundary, from the inside and from the outside, and show both scale predictably with the shape's volume. The result extends a known two-dimensional fact to all dimensions and all interpolation parameters p.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The lower-bound proof of Theorem 3.5 uses the integral formula (2.1) with the origin at 0 for the body K∩R B_2^n, but that body's centroid need not be 0; after the required recentering the support factor becomes R−⟨g,θ⟩, not R, and no bound for this factor is supplied.","rationale":"The reader's primary concern was the dependence of the lower bound on the Guedon-Milman thin-shell estimate. That estimate is a standard theorem with explicit constants, and the step from (4.3) to (4.5) is standard; I do not think the thin-shell concentration is the weakest point. The truly load-bearing issue is internal to the proof: the subset C=K∩R B_2^n is not shown to have its center of gravity at the origin, and the paper's definition of as_p requires the origin to be the centroid. The proof computes the integrand with ⟨x,N⟩=R on the spherical cap, which is only correct if the centroid of the cap equals the origin. For an asymmetric body, the centroid g of C is nonzero, and the translated boundary points are Rθ−g. This changes the support term to R−⟨g,θ⟩. For p>1 the exponent is positive, so the uncentered calculation overestimates the integral; for p<1 it underestimates, so the favorable case is not relevant. No argument is given that R−⟨g,θ⟩ stays comparable to R on a large subset of O. The isotropic condition only gives an L² bound that allows ‖g‖ of order R, and the exponent can grow with n when p is a fixed fraction of n, making even a bounded relative deficit exponentially costly. This is a correctness gap in the main lower bound, not a typo. The verdict should remain conditional: the claim may be true, but the printed proof of Theorem 3.5 needs an additional argument or a different construction. The sign error in (2.1) and the algebraic slip in combining exponents are real but secondary; the algebraic slip actually reflects the correct sign once the definition is fixed, and both are routine to repair. The centroid issue is not repaired by changing a sign.","tokens_in":17922,"tokens_out":34033,"duration_ms":330903,"concrete_test":"Take K to be the regular n-simplex in isotropic position for n=20, 50, 100 and p=n/2. Choose R from the thin-shell annulus construction, e.g. R = L_K(n^{1/2} − c n^{1/3}) with the constant from (4.3). Compute g = centroid(K∩R B_2^n), the set O = {θ : ρ_K(θ) > R}, and the corrected cap integral I = ∫_O (R − ⟨g,θ⟩)^{n(p−1)/(n+p)} dσ(θ). Compare I with c R^{n(p−1)/(n+p)} σ(O) for an absolute constant c. If the ratio tends to 0 with n, the lower-bound proof fails in this regime and needs a new argument.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 2 defines as_p(K) only after translating K so its center of gravity is at the origin. In the proof of Theorem 3.5, K is put in isotropic position (centroid 0), but the candidate subset C=K∩R B_2^n is not necessarily centered: the intersection of an asymmetric centered body with a centered ball can have nonzero centroid g. The proof then writes as_p(C) ≥ ∫_{RO} R^{-(n-1)p/(n+p)} R^{n(p-1)/(n+p)} dμ, using ⟨x,N⟩=R on the spherical cap. The correct quantity is as_p(C)=as_p(C−g), whose integrand on the corresponding cap is (R−⟨g,θ⟩)^{n(p−1)/(n+p)} with y=Rθ−g. For p>1 the exponent is positive, so replacing R−⟨g,θ⟩ by R can only overestimate the integral; a missing lower bound on R−⟨g,θ⟩ over a set of θ of measure comparable to O is needed. The isotropic condition gives only ∫_K ‖x‖²dx=nL_K², which bounds ‖g‖ ≤ C√n L_K/|C|, of the same order as R, not small. For p proportional to n, even a constant relative deficit in 1−⟨g,θ⟩/R costs an exponential factor. Thus the printed lower-bound derivation is incomplete; the sign error in (2.1) and the algebraic slip in the product are separate and fixable, but this gap concerns the main argument.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":18240,"tokens_out":23212,"duration_ms":717628,"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":[{"comment":"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":"Section 2, Eq. (2.1)"},{"comment":"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":"Section 4, proof of Theorem 3.5"},{"comment":"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.","section":"Section 4, proof of Theorem 3.5"}],"minor_comments":[{"comment":"There are two parts labelled '(i)'; the second should be labelled '(ii)'.","section":"Theorem 3.6"},{"comment":"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":"Lemma 3.2"},{"comment":"The two families of convex sets are both denoted by KK; distinct symbols, for instance ℒ_K and ℒ^K, would avoid ambiguity.","section":"Section 2"},{"comment":"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.","section":"Proposition 3.7"}],"recommendation":"major_revision","confidential_remarks":"The paper has a promising structure and the main inequalities are plausible, but the sign error in (2.1) and the missing centering argument in the lower-bound proof are not merely cosmetic. I would send this back for a careful revision rather than reject, because the upper bounds and equality cases are clean and the lower-bound construction may be repairable with a more careful choice of the subset or an additional estimate for its centroid."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat'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.\n\nNow 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.\n\nThe 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.\n\nI 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.\n\nFor 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.","headline":"Good ideas, likely true results, but the main lower-bound proof has a genuine recentering gap that must be fixed.","tokens_in":18750,"tokens_out":11991,"would_cite":false,"duration_ms":174422,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A20","52A23","52A40"],"pacs":[],"model":"deepseek-v4-flash","headline":"For convex bodies, extremal affine surface area tracks a power of volume, up to dimension-dependent constants.","keywords":["L_p affine surface area","convex bodies","extremal bodies","John ellipsoid","Lowner ellipsoid","thin shell estimate","isotropic constant","affine isoperimetric inequality"],"falsifier":"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.","tokens_in":17721,"feed_emoji":"📐","tokens_out":10065,"duration_ms":86855,"temperature":0.7,"pith_summary":"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 $-n<p\\le 0$. Since $L_p$-affine surface area measures boundary curvature in an affine-invariant way, this gives a constrained analogue of the affine isoperimetric problem with a clean asymptotic answer. The upper bound is sharp, with equality exactly for centered ellipsoids.","feed_headline":"Extremal affine surface area is a power of volume","feed_subtitle":"The largest and smallest L_p-affine surface areas inside or around a convex body are set by volume, not shape.","key_machinery":"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.","core_discovery":"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 $n^{n(n-p)/(2(n+p))}$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the thin-shell concentration estimate that yields a half-volume shell of width $O(n^{1/3})$ around $L_K\\sqrt{n}$, the key to the lower bound.","marker":"[23]"},{"why":"Introduces $L_p$-affine surface area and proves the $L_p$-affine isoperimetric inequality and upper semicontinuity used in the upper bound.","marker":"[38]"},{"why":"Extends the $L_p$-affine isoperimetric inequalities to all $p$ and supplies the monotonicity of normalized affine surface areas used in Proposition 3.4.","marker":"[60]"},{"why":"Determines the planar $p=1$ extremal inner body and motivates the higher-dimensional question.","marker":"[3]"},{"why":"Provides the volume product lower bound $|L||L^\\circ|\\ge c^n|B_2^n|^2$ used in the outer lower estimate.","marker":"[11]"},{"why":"F. John's theorem bounds an affine image of any containing body between $rB_2^n$ and $nrB_2^n$, used to restrict the bodies competing for the outer quantities.","marker":"[32]"},{"why":"Supplies the Lowner ellipsoid inclusions used to turn the outer estimates into powers of volume.","marker":"[26]"}],"fun_headline_variants":["Volume determines extremal affine surface area","Extremal affine areas scale as volume power","Ball and ellipsoids set extreme affine areas","Affine area extrema pinned to volume alone","Volume power controls affine surface area extremes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Volume determines extremal affine surface area","Extremal affine areas scale as volume power","Ball and ellipsoids set extreme affine areas","Affine area extrema pinned to volume alone","Volume power controls affine surface area extremes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000211,"raw_usage":{"total_tokens":1450,"prompt_tokens":1014,"completion_tokens":436,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":630,"completion_tokens_details":{"reasoning_tokens":369}},"tokens_in":630,"tokens_out":436,"duration_ms":4932,"temperature":1.0,"reasoning_tokens":369,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:55:24.267118+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Gu´ edon and E","cited_arxiv_id":null,"evidence_quote":"Supplies the thin-shell concentration estimate that yields a half-volume shell of width $O(n^{1/3})$ around $L_K\\sqrt{n}$, the key to the lower bound."},{"cited_title":"Lutwak, The Brunn-Minkowski-Firey theory","cited_arxiv_id":null,"evidence_quote":"Introduces $L_p$-affine surface area and proves the $L_p$-affine isoperimetric inequality and upper semicontinuity used in the upper bound."},{"cited_title":"Werner and D","cited_arxiv_id":null,"evidence_quote":"Extends the $L_p$-affine isoperimetric inequalities to all $p$ and supplies the monotonicity of normalized affine surface areas used in Proposition 3.4."},{"cited_title":"B´ ar´ any,Affine perimeter and limit shape , J","cited_arxiv_id":null,"evidence_quote":"Determines the planar $p=1$ extremal inner body and motivates the higher-dimensional question."},{"cited_title":"Bourgain, V","cited_arxiv_id":null,"evidence_quote":"Provides the volume product lower bound $|L||L^\\circ|\\ge c^n|B_2^n|^2$ used in the outer lower estimate."},{"cited_title":"John, Extremum problems with inequalities as subsidiary conditi ons, Courant Anniversary Volume, Interscience, New York (1948) 187–204","cited_arxiv_id":null,"evidence_quote":"F. John's theorem bounds an affine image of any containing body between $rB_2^n$ and $nrB_2^n$, used to restrict the bodies competing for the outer quantities."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Lowner ellipsoid inclusions used to turn the outer estimates into powers of volume."}],"review_version":1}