{"id":"c94b78a9-f35e-48ad-adbe-42eb91aeb883","arxiv_id":"2412.16054","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For fixed m, the rescaled volume of a random m-dimensional projection or section of an ell_p ball satisfies a CLT, an MDP, and an LDP with explicit limits as N grows.","lead":"This paper proves central limit theorems, moderate deviation principles, and large deviation principles for the volume of random projections and sections of high-dimensional ell_p balls. The explicit constants and rate functions complete the fluctuation picture for these basic convex bodies and refine a decade-old theorem on projections of the cube.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem C section LDP rate function uses exponent q>2, outside the Stiefel LDP range; sections require exponent p.","rationale":"The reader's verdict is CONDITIONAL, focusing on the black-box Stiefel LDP and the p=1 section case of Theorem A. My stress-test identifies a more concrete, internal problem in Theorem C: the section LDP rate function is stated in terms of q, the Hölder conjugate of p, but for sections p∈[1,2) implies q>2, outside the range where the functional LDP (Theorem 2.4) and the underlying Sanov theorem (Lemma 4.13) are established. The proof of Theorem C for sections must use the p-th power representation from Lemma 4.4, so the rate function should involve p, not q. This is a load-bearing flaw in the statement of a central theorem, but it is local and fixable by replacing q with p in the definition of I'_LDP for the section case. The CLT (Theorem A) and MDP (Theorem B) appear sound aside from the minor p=1 coverage gap in Lemma 4.4, which is trivially patched. Since the verdict CONDITIONAL already requires corrections, my concern does not change the verdict; it sharpens the conditions. Agreement with the reader is partial: both flag the LDP machinery, but the reader attributes risk to the imported Stiefel result, whereas the sharper risk is the exponent mismatch inside Theorem C.","tokens_in":36273,"tokens_out":19144,"duration_ms":152578,"concrete_test":"Specialize to m=1, p=1 (random diameter of the rescaled cross-polytope). Derive the LDP rate function for vol_1(N^{-1/2} B_1^N ∩ E_N) by applying Theorem 2.4 with exponent p=1 and the continuous transformation in Lemma 4.4, and compare it with the formula in Theorem C using q=∞. If the two rate functions differ, the stated section rate function is wrong. More generally, re-derive the section case of Theorem C by applying the contraction principle to X_N^{(p)} = (1/N)∑|⟨√N v_i,·⟩|^p and check whether I'_LDP must be defined with p instead of q.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Theorem C, the section rate function I^∩_LDP is defined via I'_LDP(f) = sup_k inf { I_Stiefel(µ) : (f^q(u_1),...,f^q(u_k)) = F_k(µ) }, where q is the Hölder conjugate of p. For ♦=∩ the allowed range is p∈[1,2), hence q∈(2,∞]. But the functional LDP (Theorem 2.4) and the Stiefel Sanov lemma (Lemma 4.13) are proved only for exponents in (0,2); Theorem 2.4 relies on exponential tightness of q-th powers of N(0,Id_m), which fails for q>2. Moreover, the support-function representation for sections (Lemma 4.4) expresses the volume as κ_m ∫ ((1/N)∑|⟨√N v_i,u⟩|^p)^{-m/p} σ(du), i.e., with the p-th power, not the q-th. Therefore the contraction principle can only produce an LDP whose rate function is defined via p for the section case. As written, the explicit rate function for ♦=∩ in Theorem C is not justified and likely incorrect, while the projection case (q∈[1,2)) is fine. This is an internal inconsistency in a main theorem, distinct from the black-box reliance on [52].","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies fixed-dimensional random projections and sections of ℓ_p^N balls. For p∈[1,∞] and fixed m, it proves a CLT at the N^{1/2} scale (Theorem A), an MDP (Theorem B), and an LDP (Theorem C) for the volume, with explicit means, variances, speeds, and rate functions. These results are derived from functional CLT/MDP/LDP theorems for processes of the form u ↦ (1/N)∑_{i=1}^N |⟨√N v_i,u⟩|^q, where v_i are columns of a uniform Stiefel matrix (Theorems 2.2–2.4), followed by Hadamard-differentiable maps to the volume and contraction/delta-method arguments. The paper also proves almost-sure approximation of the projected/sectioned bodies by Euclidean balls.","tokens_in":36454,"tokens_out":20392,"duration_ms":171397,"significance":"If the main theorems hold, the paper provides a complete asymptotic picture for fixed-dimensional random projections and sections of ℓ_p balls, generalizing the cube-projection CLT of Paouris–Pivovarov–Zinn and the section results of Adamczak–Pivovarov–Simanjuntak, and giving explicit non-universal rate functions. The proof strategy is coherent and the paper contains detailed moment computations (Lemmas 4.7–4.9) that support the explicit constants. There are no fitted parameters. However, the LDP for sections has an exponent mismatch that affects a main theorem, and the statement and proof of the functional LDP Theorem 2.4 are inconsistent; these issues are locally fixable but must be corrected.","major_comments":[{"comment":"The LDP for sections in Theorem C is not justified as stated. For ♦=∩ the paper takes p∈[1,2), so the Hölder conjugate q=p/(p−1) lies in (2,∞], while the functional LDP (Theorem 2.4) and the Sanov-type lemma (Lemma 4.13) are proved only for exponents in [1,2) (Lemma 4.13 states 0<q<2). Moreover, Lemma 4.4 represents the section volume through the p-th power average X_N(u)=(1/N)∑_{i=1}^N |⟨√N v_i,u⟩|^p, not through the q-th power. Therefore the contraction principle can only produce a section LDP whose rate function is defined via F^p_k(µ)=∫|<x,u>|^p dµ, not via the displayed F_k with exponent q. The proof in Section 4.7 does not address this discrepancy, so the section case of Theorem C is internally inconsistent as written.","section":"Theorem C (Section 2.1); proof in Section 4.7"},{"comment":"The statement of Theorem 2.4 and its proof disagree on the rate function. The theorem states I'_LDP(f)=sup_k inf{I_Stiefel(µ): (f^q(u1),...,f^q(uk))=F_k(µ)}, but the proof in Section 4.6 derives the rate with the condition F^(k)(µ)=(f(u1),...,f(uk)). Since X_N(u_i)=∫|<x,u_i>|^q dL_N(x)=F^(k)(L_N)_i, the proof's version is the correct rate for the LDP of X_N. As printed, the theorem's rate function is the rate for the q-th root of X_N, not for X_N itself. This inconsistency propagates to Theorem C: in the projection case, h^q=X_N, so under the proof's convention the contraction should use I'_LDP(h^q) rather than I'_LDP(h). The authors should adopt one convention consistently in the statement, proof, and applications.","section":"Theorem 2.4 (statement vs. proof in Section 4.6)"},{"comment":"Theorem A explicitly extends the section case to p=1, but the proof does not cover this value. The volume representation in Lemma 4.4 is stated only for p∈(1,∞], and the proof of Theorem A in Section 4.7 invokes this lemma for the section case. Since p=1 is included in Theorem A and also in Theorem C for sections, either Lemma 4.4 must be extended to p=1 (the argument appears to work with minor changes, using Theorem 2.2 with exponent p=1 rather than q=∞) or a separate proof must be supplied. As written, this is a gap in a stated part of a main theorem.","section":"Theorem A; Lemma 4.4; Theorem C"}],"minor_comments":[{"comment":"The section heading contains a typo: 'Brackground' should be 'Background'.","section":"Section 3.3"},{"comment":"'Its is immediate from the discussion above' should read 'It is immediate from the discussion above'.","section":"Introduction, paragraph 3"},{"comment":"The phrase 'the symbol stands for for an orthogonal projection or a section' contains a duplicated 'for'.","section":"Section 2.1, definition of ♦"},{"comment":"The constraint in the section case is written as ∫_{S^{m-1}} f^{-k/p}(t)dt = x, but the dimension parameter k is not defined there; it should presumably be m, so that the integral matches the representation in Lemma 4.4.","section":"Theorem C, definition of I^∩_LDP"},{"comment":"The typeset formula for σ^2_∩ appears garbled, particularly the expression 'mπ(pm +m)/p'; please check the exponent and typeset the formula so that it matches the derivation at the end of the proof of Theorem A.","section":"Remark 2.1, formula for σ^2_∩"},{"comment":"In the proof of Theorem 2.4, the displayed definition of F^(k)(µ) has |⟨√N v_i,u_j⟩|^q inside the integral with respect to µ(dx); it should be |⟨x,u_j⟩|^q.","section":"Proof of Theorem 2.4, definition of F^(k)"},{"comment":"In the last display of the proof of Lemma 4.10, the factor ∥x_i∥^q should be ∥x_i∥^{q-1} to match the Lipschitz estimate in the statement.","section":"Lemma 4.10"}],"recommendation":"major_revision","confidential_remarks":"The paper is largely built on the Sanov-type LDP of Kim–Ramanan [52, Theorem 2.8], which is imported as a black box. Assuming that result is correct, the main obstacles are the section LDP exponent mismatch in Theorem C and the inconsistency between the statement and proof of Theorem 2.4; both appear fixable without changing the overall approach. I therefore recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a serious paper and the projection side is in good shape, but the theorem as written overreaches on sections. The section case of Theorem C defines its rate function through the Hölder conjugate q, while the volume representation in Lemma 4.4 and the contraction step in the proof use the p-th power. For sections p∈[1,2), so q>2, and the Stiefel LDP from [52] is only valid for exponents below 2. Thus the stated rate function for sections is not justified and, as far as I can tell, not correct as written. That is not a cosmetic issue: it is a main theorem. The fix is probably straightforward—redefine I'_{LDP} for sections using p-th powers—but it has to be done and checked.\n\nWhat is genuinely new: exact variance constants for the cube projection (previously unknown), the CLT for general p with explicit μ and σ², the MDP, and the LDP for projections. The functional framework over the Stiefel manifold is clearly laid out, the moment computations are thorough, and the paper is honest about importing the Sanov-type LDP from [52]. The p=∞ projection case and p∈(1,2) section CLT look convincing to me.\n\nSoft spots: (1) The section LDP exponent problem above. (2) Theorem A states p=1 for sections, but Lemma 4.4 is proved only for p>1, and no limiting argument is given. This is a smaller gap, likely fixable. (3) The reliance on [52] is real but not a flaw—the cited theorem is the right tool, and the paper identifies exactly what it needs. (4) I haven't machine-checked the long covariance computations; they look plausible and the back-of-envelope consistency checks pass.\n\nWho this is for: specialists in asymptotic geometric analysis and high-dimensional probability. It deserves referee time, but a serious referee should insist on fixing the section LDP before publication. I'd cite the projection results and the CLT constants, not the section LDP as stated.","headline":"A rich, mostly sound paper whose stated section LDP rate function is undercut by an exponent mismatch, and whose p=1 CLT is announced but not proved.","tokens_in":37061,"tokens_out":3906,"would_cite":true,"duration_ms":30538,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A23","60F05","60F10","46B09","52A22","60D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every fixed dimension m and every p in (1,∞)\\textbackslash{2}, the volume of a uniformly random m-dimensional projection or section of a rescaled $\\ell_p^N$-ball fluctuates around an explicit constant like a Gaussian at scale…","keywords":["asymptotic theory of convex bodies","central limit theorem","large deviation principle","moderate deviation principle","\\ell_p-balls","random projection","random section","stochastic geometry"],"falsifier":"Take $m=2$, $p=3$, sample many uniform 2-dimensional subspaces of $\\mathbb{R}^N$ for very large $N$, and compare the empirical distribution of $\\sqrt{N}(\\mathrm{vol}_2(N^{1/3-1/2}(B_3^N|E_N))-\\mu_\\diamond)$ with $\\mathcal{N}(0,\\sigma_\\diamond^2)$; a mismatch at the $1/\\sqrt{N}$ scale would disprove the CLT.","tokens_in":36006,"feed_emoji":"🎲","tokens_out":12247,"duration_ms":98790,"temperature":0.7,"pith_summary":"The paper proves that random orthogonal projections and sections of high-dimensional $\\ell_p$-balls, cut by a uniformly random subspace of fixed dimension $m$, are asymptotically normal after the natural rescaling $N^{1/p-1/2}$. The limiting mean and variance are given by closed-form expressions in $m$ and $p$, and the coverage extends the previously known cube and hyperplane-section central limit theorems to the full range $p\\in(1,\\infty)\\setminus\\{2\\}$ (with $p=\\infty$ for projections and $p=1$ for sections). The same framework produces moderate and large deviation principles with explicit speeds and rate functions, so the rare-event behaviour is shown to depend on the geometry of $\\ell_p$ rather than being universal. A sympathetic reader would care because these are the first results that give a complete asymptotic picture, fluctuation scale, moderate deviations, and large deviations, for fixed-dimensional random shadows and slices of the standard family of high-dimensional convex bodies.","feed_headline":"Random slices and shadows of ℓp-balls turn Gaussian at 1/√N","feed_subtitle":"For each fixed dimension, projections and sections of ℓp-balls follow an explicit CLT—with moderate and large deviations too.","key_machinery":"The argument runs through the Stiefel manifold: a uniform $m$-subspace is the row space of a uniform element $V_N\\in V_{m,N}$, and the support function of the projected ball satisfies $h(N^{1/2-1/q}V_N B_p^N,u)^q = \\frac{1}{N}\\sum_{i=1}^N |\\langle\\sqrt{N}v_i,u\\rangle|^q$, where $v_i$ are the columns of $V_N$ and $q$ is the H\\\"older conjugate of $p$. Section volumes have an analogous representation with $p$ in place of $q$, integrated against the normalized spherical measure. A functional central limit theorem for these sums on $S^{m-1}$ is obtained by a Taylor expansion that separates the Gaussian matrix from its inverse square root, producing a Gaussian process with a covariance kernel computed in closed form from absolute moments of inner products of standard Gaussian vectors. Hadamard differentiability of the maps $f\\mapsto f^\\alpha$ and of the radial-function-to-volume map transfers the functional limits to the volume, while the large deviation principle is built from a Sanov-type large deviation principle for the empirical measure of the scaled columns $\\sqrt{N}v_i$ together with the contraction principle.","core_discovery":"The central discovery is a fluctuation theorem: for fixed $m$ and $p\\in(1,\\infty)\\setminus\\{2\\}$, the volume of a uniformly random $m$-dimensional projection or section of the rescaled ball $N^{1/p-1/2}B_p^N$ converges in distribution, after centering by an explicit $\\mu_\\diamond$ and scaling by $\\sqrt{N}$, to a normal distribution with explicit variance $\\sigma_\\diamond^2$ (given in Remark 2.1). Projections include $p=\\infty$ and sections include $p=1$, recovering the cube CLT with explicit normalizing constants and generalizing the hyperplane-section CLT to fixed codimension. Beyond the central limit, the paper proves a moderate deviation principle at intermediate scales and a large deviation principle at speed $N$, with rate functions expressed through a relative-entropy-plus-covariance functional on probability measures on $\\mathbb{R}^m$. Together these form a complete asymptotic description of the volume of random projections and sections of $\\ell_p^N$-balls.","pith_inferences":["A consequence the paper leaves implicit: as the subspace dimension $m$ grows with $N$, the variance formulas degenerate in some parameter ranges, so the $\\sqrt{N}$ scaling should eventually give way to a different speed; locating that transition is a natural next step.","The same support-function representation may transfer to other convex bodies whose dual norm is a sum of one-dimensional contributions, such as Orlicz balls, yielding Gaussian fluctuations with covariance built from the same absolute-moment integrals.","The large deviation rate function suggests a maximum-entropy picture: rare volumes are achieved by an empirical distribution of the random matrix columns that minimizes relative entropy plus a covariance penalty; this could be tested by conditionally sampling Stiefel columns under a volume constraint.","A concrete simulation check would be to take $p=3$, $m=2$, generate many random 2-subspaces in $\\mathbb{R}^N$ for large $N$, and compare the histogram of $\\sqrt{N}(\\mathrm{vol}-\\mu_\\diamond)$ with the predicted normal density; the fit should hold to Monte Carlo error."],"forward_implications":["For fixed $m$ and $p\\in(1,\\infty)\\setminus\\{2\\}$, volumes of random projections and sections of $N^{1/p-1/2}B_p^N$ fluctuate as $\\mu_\\diamond \\pm \\sigma_\\diamond/\\sqrt{N}$ to first order in distribution.","The cube case $p=\\infty$ (projections) and cross-polytope case $p=1$ (sections) are covered, so the known cube CLT is recovered with explicit variance.","At intermediate scales $\\beta_N\\to\\infty$, $\\beta_N=o(\\sqrt{N})$, the same fluctuations satisfy a moderate deviation principle with speed $\\beta_N^2$ and a quadratic rate function, so Gaussian tails persist until $\\beta_N$ reaches the $\\sqrt{N}$ scale.","At speed $N$, the large deviation rate function is expressed as an infimum over probability measures on $\\mathbb{R}^m$, so rare events are governed by a relative-entropy-plus-covariance criterion rather than by a universal law.","Since the limit theorems hold at the level of random functions on the sphere, continuous functionals of the support or radial function---including intrinsic volumes and dual volumes---inherit the corresponding CLT, MDP, and LDP."],"supporting_citations":[{"why":"Supplies the Sanov-type large deviation principle for the empirical measure of Stiefel columns that the paper imports as Lemma 4.13 and uses to build the volume LDP.","marker":"[52]"},{"why":"The previously known central limit theorem for random projections of the cube, which the paper generalises to all $\\ell_p$-balls and complements with explicit mean and variance.","marker":"[66]"},{"why":"The central limit theorem for small-codimensional random sections of $\\ell_p$-balls that the paper extends to fixed dimension and rounds out with MDP and LDP statements.","marker":"[2]"},{"why":"Provides the functional central limit theorem for sample-continuous processes used to prove the functional CLT on the sphere (Proposition 3.12).","marker":"[30]"},{"why":"Supplies the contraction principle, exponential equivalence, and exponential tightness results used throughout the LDP and MDP proofs.","marker":"[20]"},{"why":"Gives the delta method for large deviations used to transfer functional MDP/LDP statements to the volume.","marker":"[29]"},{"why":"Gives the infinite-dimensional delta method used to transfer the functional CLT to the volume in Theorem A.","marker":"[70]"},{"why":"Supplies the Dawson-G\\\"artner projective limit theorem used to turn finite-dimensional evaluation LDPs into a weak LDP on the function space.","marker":"[18]"}],"fun_headline_variants":["ℓp-ball random slices and shadows turn Gaussian at 1/√N","Explicit CLT for volume of random ℓp-ball projections and sections","CLT, moderate and large deviations for ℓp-ball section volumes","Volume fluctuations of ℓp-ball sections and projections: Gaussian limit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire large-deviation result rests on an imported theorem about how the empirical distribution of the columns of a random orthogonal-frame matrix concentrates, and the $p=1$ section case of the central limit theorem is stated without proof; if either gap is real, the corresponding part of the claimed picture collapses.","fun_headline_variants_meta":{"raw":{"variants":["ℓp-ball random slices and shadows turn Gaussian at 1/√N","Explicit CLT for volume of random ℓp-ball projections and sections","CLT, moderate and large deviations for ℓp-ball section volumes","Volume fluctuations of ℓp-ball sections and projections: Gaussian limit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000805,"raw_usage":{"total_tokens":3557,"prompt_tokens":985,"completion_tokens":2572,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":601,"completion_tokens_details":{"reasoning_tokens":2492}},"tokens_in":601,"tokens_out":2572,"duration_ms":18951,"temperature":1.0,"reasoning_tokens":2492,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:51:11.892180+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $m=2$, $p=3$, sample many uniform 2-dimensional subspaces of $\\mathbb{R}^N$ for very large $N$, and compare the empirical distribution of $\\sqrt{N}(\\mathrm{vol}_2(N^{1/3-1/2}(B_3^N|E_N))-\\mu_\\diamond)$ with $\\mathcal{N}(0,\\sigma_\\diamond^2)$; a mismatch at the $1/\\sqrt{N}$ scale would disprove the CLT.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Sanov-type large deviation principle for the empirical measure of Stiefel columns that the paper imports as Lemma 4.13 and uses to build the volume LDP."},{"cited_title":"Paouris, P","cited_arxiv_id":null,"evidence_quote":"The previously known central limit theorem for random projections of the cube, which the paper generalises to all $\\ell_p$-balls and complements with explicit mean and variance."},{"cited_title":"Adamczak, P","cited_arxiv_id":null,"evidence_quote":"The central limit theorem for small-codimensional random sections of $\\ell_p$-balls that the paper extends to fixed dimension and rounds out with MDP and LDP statements."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the functional central limit theorem for sample-continuous processes used to prove the functional CLT on the sphere (Proposition 3.12)."},{"cited_title":"Dembo and O","cited_arxiv_id":null,"evidence_quote":"Supplies the contraction principle, exponential equivalence, and exponential tightness results used throughout the LDP and MDP proofs."},{"cited_title":"Gao and X","cited_arxiv_id":null,"evidence_quote":"Gives the delta method for large deviations used to transfer functional MDP/LDP statements to the volume."},{"cited_title":"Roemisch","cited_arxiv_id":null,"evidence_quote":"Gives the infinite-dimensional delta method used to transfer the functional CLT to the volume in Theorem A."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Dawson-G\\\"artner projective limit theorem used to turn finite-dimensional evaluation LDPs into a weak LDP on the function space."}],"review_version":1}