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Limit Theorems for the Volume of Random Projections and Sections of $\ell_p^N$-balls

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2412.16054 v1 pith:2BPHMNGS submitted 2024-12-20 math.PR math.MG

Joscha Prochno , Christoph Thaele , Philipp Tuchel This is my paper · ORCID
classification math.PRmath.MG MSC 52A2360F0560F1046B0952A2260D05
keywords asymptotictheoryofconvexbodiescentrallimittheoremlargedeviationprinciplemoderate\ell_p-ballsrandomprojectionsectionstochasticgeometry
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

3 major / 7 minor

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.

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 (3)
  1. [Theorem C (Section 2.1); proof in Section 4.7] 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.
  2. [Theorem 2.4 (statement vs. proof in Section 4.6)] 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.
  3. [Theorem A; Lemma 4.4; Theorem C] 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.
minor comments (7)
  1. [Section 3.3] The section heading contains a typo: 'Brackground' should be 'Background'.
  2. [Introduction, paragraph 3] 'Its is immediate from the discussion above' should read 'It is immediate from the discussion above'.
  3. [Section 2.1, definition of ♦] The phrase 'the symbol stands for for an orthogonal projection or a section' contains a duplicated 'for'.
  4. [Theorem C, definition of I^∩_LDP] 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.
  5. [Remark 2.1, formula for σ^2_∩] 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.
  6. [Proof of Theorem 2.4, definition of F^(k)] 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.
  7. [Lemma 4.10] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the constants and rate functions are computed from stated integrals and an external Sanov-type theorem, not fitted or self-referential.

full rationale

The paper's derivation chain is self-contained against external benchmarks. The CLT and MDP proofs build functional limit theorems for sums over Stiefel-manifold columns via a Gaussian matrix decomposition, a Taylor expansion, and the delta method, with the asymptotic mean and variance evaluated from explicit moment integrals in Lemmas 4.7-4.9; no parameter is fitted to data and no quantity is defined in terms of the target limit. The LDP imports the Sanov-type empirical-measure LDP from Kim and Ramanan [52, Theorem 2.8] as an external theorem, the authors of which do not overlap with the present paper, and the rate function I_Stiefel is not derived from the volume result being proved. The self-citations in the paper are contextual or complementary and are not load-bearing for the main derivations, which are proved in the text. The reviewer-flagged concern that the section-case LDP in Theorem C uses an exponent q>2 outside the range of the imported Stiefel LDP is a correctness or foundational issue about the applicability of an external result, not a circularity: it does not make the conclusion equivalent to its input by construction. Similarly, the black-box reliance on [52] is a support risk, not a circular step.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no free parameters, no fitted constants, and no new entities. The explicit mu and sigma are derived, not fitted. The main external inputs are standard theorems in probability and convex geometry plus the Stiefel empirical-measure LDP from [52]. There is no circularity in the sense of fitting or normalizing to force the result.

assumptions (3)
  • domain assumption Sanov-type LDP for empirical measures of columns of random Stiefel matrices (Lemma 4.13, cited from [52, Theorem 2.8]) with rate I_Stiefel(nu) = H(nu|gamma^otimes m) + (1/2) Trace(Id_m - C(nu)).
    Used as the input for the functional LDP and hence for the volume LDP; no proof is given in this paper.
  • standard math Gaussian representation of uniform Stiefel columns: v_i = (G_N G_N^*)^{-1/2} g_i for a standard Gaussian matrix G_N.
    Used in Section 4.5 to reduce Stiefel sums to i.i.d. Gaussian functionals.
  • standard math Volume representation of a convex body from its radial function: vol_m(C) = kappa_m integral over S^{m-1} of rho(C,u)^m sigma(du).
    Used in Lemmas 4.4 and 4.6 and throughout Section 4.7 to pass from support and radial functions to volume.

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Cite this review

Pith. "Pith review of Limit Theorems for the Volume of Random Projections and Sections of $\ell_p^N$-balls." pith.science (2026). https://pith.science/paper/2BPHMNGS

@misc{pith2026241216054,
  author       = {Pith},
  title        = {Pith review of: Limit Theorems for the Volume of Random Projections and Sections of $\ell_p^N$-balls},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2BPHMNGS}},
  note         = {Machine review of arXiv:2412.16054}
}
abstract

Let $\mathbb{B}_p^N$ be the $N$-dimensional unit ball corresponding to the $\ell_p$-norm. For each $N\in\mathbb N$ we sample a uniform random subspace $E_N$ of fixed dimension $m\in\mathbb{N}$ and consider the volume of $\mathbb{B}_p^N$ projected onto $E_N$ or intersected with $E_N$. We also consider geometric quantities other than the volume such as the intrinsic volumes or the dual volumes. In this setting we prove central limit theorems, moderate deviation principles, and large deviation principles as $N\to\infty$. Our results provide a complete asymptotic picture. In particular, they generalize and complement a result of Paouris, Pivovarov, and Zinn [A central limit theorem for projections of the cube, Probab. Theory Related Fields. 159 (2014), 701-719] and another result of Adamczak, Pivovarov, and Simanjuntak [Limit theorems for the volumes of small codimensional random sections of $\ell_p^n$-balls, Ann. Probab. 52 (2024), 93-126].

Figures

Figures reproduced from arXiv: 2412.16054 by the authors.

Figure 1
Figure 1. Plot of the asymptotic variance σ 2 ♦ in the case of projections in the parameter q for different values of m [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Plot of the asymptotic variance σ 2 ♦ in the case of sections in the parameter p for different values of m. Clearly the plots in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Plot of the asymptotic variance σ 2 ♦ in the case of projections in the parameter m for different values of q. On a formal level a large deviation principle (LDP) is the same as a moderate deviation principle as in (1), but with βN = p N. To present our third main result, an LDP for the volume of projections and sections of ℓp-balls, let us define Fk (µ) := ³ Z Rm |〈x,u1〉|q dµ(x) ´ 1 q ,...,Z Rm |〈x,uk 〉|q dµ(x) ´ 1… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Plot of the asymptotic variance σ 2 ♦ in the case of sections in the parameter m for different values of p. for its m-fold product measure on R m. For µ,ν ∈ M1(R m), the relative entropy or Kullback–Leibler divergence H(ν|µ) is defined as H(ν|µ) := (R Rm log³ dν dµ ´ d…

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Cited by 1 Pith paper

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  1. Strange shadows of $\ell_p$-balls

    math.PR 2024-12 conditional novelty 8.0 of 10

    Random projections of ℓ_p-balls satisfy a large deviations principle whose rate function is finite only on L_q-zonoids and given by a maximum entropy gap, with almost sure convergence to a Euclidean ball.

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Pith tools

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