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arxiv: 2605.11995 · v1 · submitted 2026-05-12 · 🧮 math.PR · math.FA· math.MG

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· Lean Theorem

Intrinsic volumes of ell_p-balls and a continuum of Maxwell--Poincar\'e--Borel laws for their curvature measures

Alexander Marynych, Joscha Prochno, Zakhar Kabluchko

Pith reviewed 2026-05-13 04:50 UTC · model grok-4.3

classification 🧮 math.PR math.FAmath.MG
keywords intrinsic volumesℓ_p-ballscurvature measuresMaxwell-Poincaré-Borel lawshigh-dimensional limitsconvex geometrymixed moments
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The pith

Intrinsic volumes of ℓ_p-balls admit explicit one-dimensional integral formulas, and their curvature measures obey a family of Maxwell-Poincaré-Borel limit laws indexed by α in [0,1].

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives closed-form expressions for all intrinsic volumes of the n-dimensional ℓ_p-ball and its coordinate-weighted versions when p exceeds 1. These volumes are written as a single integral against the auxiliary function F_p that integrates |u| to a power times an exponential decay involving both p and 2p-2. The same machinery produces a high-dimensional limit theorem: when a boundary point is drawn from the normalized j-th curvature measure with j/n approaching α, the first r scaled coordinates converge in law to a product of r independent copies of an explicit probability measure ν_{p,α}. A sympathetic reader cares because the formulas recover every previously known special case for ellipsoids, crosspolytopes, and boxes while turning high-dimensional curvature into an object that can be studied coordinate-wise.

Core claim

For p>1 the intrinsic volumes V_j(B_p^n) are given explicitly by a one-dimensional integral involving the special function F_p(t;ν) = ∫_R |u|^ν exp(-|u|^p - t |u|^{2p-2}) du; the same representation yields asymptotic expressions when the index j depends on n. The curvature measures Φ_j(B_p^n,·) admit an explicit characterization whose mixed moments are also closed-form, and this characterization implies that if X_n is sampled from the normalized curvature measure Φ_{j(n)} with j(n)/n → α then n^{1/p} X_n has its first r coordinates converging weakly to the product measure ν_{p,α}^⊗r for any fixed r.

What carries the argument

Explicit characterization of the curvature measures of coordinate-weighted ℓ_p-balls together with closed-form expressions for all their mixed moments.

If this is right

  • All previously known explicit formulas for intrinsic volumes of ellipsoids, weighted crosspolytopes, and rectangular boxes are recovered as special or limiting cases of the new integral expression.
  • Asymptotic formulas hold for V_{j(n)}(B_p^n) whenever j(n) is any sequence with j(n)/n tending to a fixed α in [0,1].
  • The limiting measure ν_{p,α} is an explicit probability law on R whose moments can be read off from the same integral representation used for the volumes.
  • The joint convergence holds simultaneously for any fixed number r of coordinates, giving a product structure in the limit.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The explicit integral formulas open the possibility of stable numerical quadrature for intrinsic volumes at moderate n, replacing Monte-Carlo sampling of surface measures.
  • Because the limit laws factorize across coordinates, similar product structures may appear for curvature measures of other symmetric convex bodies once their mixed-moment formulas are known.
  • The parameter α interpolates continuously between volume-type and surface-type measures, suggesting a unified way to study concentration phenomena across different codimensions.

Load-bearing premise

The curvature measures of coordinate-weighted ℓ_p-balls must possess the stated explicit moment formulas without requiring extra unstated regularity conditions.

What would settle it

For the Euclidean case p=2 and small n, compute the integral formula for V_{n-1}(B_2^n) and check whether it equals the known surface area of the unit ball; mismatch for any n>1 would falsify the general claim.

read the original abstract

For $p>1$, we derive explicit formulas for the intrinsic volumes $V_0(\mathbb B_p^n),\dots,V_{n-1}(\mathbb B_p^n)$ of the $n$-dimensional $\ell_p$-balls $$ \mathbb B_p^n = \{x\in\mathbb R^n:\ |x_1|^p+\ldots+|x_n|^p\le 1\} $$ and, more generally, of their coordinate-weighted analogues. The formula is given in terms of a one-dimensional integral involving the special function $$ \mathcal F_p(t;\nu) = \int_{\mathbb R}|u|^\nu e^{-|u|^p-t|u|^{2p-2}}\,du. $$ Previously known formulas for the intrinsic volumes of ellipsoids, weighted crosspolytopes, and rectangular boxes arise as special or limiting cases. We also obtain asymptotic formulas for $V_{j(n)}(\mathbb B_p^n)$ in the high-dimensional regime $n\to\infty$, where the index $j(n)$ is allowed to depend on $n$. We further investigate the curvature measures of $\mathbb B_p^n$. These are finite measures $$ \Phi_0(\mathbb B_p^n,\cdot),\dots,\Phi_{n-1}(\mathbb B_p^n,\cdot) $$ on $\partial\mathbb B_p^n$ that localize the intrinsic volumes. We prove a Maxwell--Poincar\'{e}--Borel type limit theorem: if $X_n$ is a random boundary point of $\mathbb B_p^n$ distributed according to the normalized curvature measure $\Phi_{j(n)}(\mathbb B_p^n,\cdot)/V_{j(n)}(\mathbb B_p^n)$, where $j(n)/n\to\alpha\in[0,1]$ as $n\to\infty$, then for every fixed $r\in\mathbb N$, the joint distribution of the first $r$ coordinates of $n^{1/p}X_n$ converges weakly to the product measure $\nu_{p,\alpha}^{\otimes r}$. Here $\nu_{p,\alpha}$ is an explicit probability measure on $\mathbb R$ depending on $p>1$ and $\alpha\in[0,1]$. The main tool underlying these results is an explicit characterization of the curvature measures of coordinate-weighted $\ell_p$-balls, and in particular an explicit formula for their mixed moments.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 4 minor

Summary. The paper derives explicit formulas for the intrinsic volumes V_0(B_p^n), …, V_{n-1}(B_p^n) of the ℓ_p-ball and its coordinate-weighted analogues for p > 1, expressed as one-dimensional integrals involving the special function F_p(t; ν) = ∫ |u|^ν exp(−|u|^p − t |u|^{2p−2}) du. These formulas recover the known cases for ellipsoids (p=2), crosspolytopes, and boxes. The authors also prove a Maxwell–Poincaré–Borel-type weak-convergence theorem: under the normalized curvature measure Φ_{j(n)}(B_p^n, ·)/V_{j(n)}(B_p^n) with j(n)/n → α ∈ [0,1], the first r coordinates of n^{1/p} X_n converge in distribution to the product measure ν_{p,α}^{⊗r}, where ν_{p,α} is an explicit probability measure on R. The proofs rest on an explicit characterization of the curvature measures of weighted ℓ_p-balls together with closed-form mixed-moment formulas.

Significance. If the derivations hold, the work supplies the first explicit integral expressions for intrinsic volumes of ℓ_p-balls in all dimensions and a continuous family of high-dimensional limit laws for their curvature measures, parameterized by α. The reduction to classical cases (ellipsoids, crosspolytopes, boxes) and the explicit moment formulas constitute verifiable strengths. These results are likely to be useful in convex geometry, asymptotic analysis of norms, and probability on convex bodies.

minor comments (4)
  1. [§1] §1 (Introduction): the statement that the formulas are 'parameter-free' should be clarified, since the integral representation still depends on the auxiliary parameter t that is later optimized or integrated out; a short remark on how this differs from truly parameter-free expressions would prevent misreading.
  2. [Definition of F_p] Definition of F_p(t;ν) (p. 3): the domain of ν is not stated explicitly. Since the intrinsic-volume formulas invoke ν = j, j+1, …, n−1, a sentence confirming that the integral converges for all real ν > −1 (or the precise range needed) would improve readability.
  3. [Limit theorem] Theorem on the limit law (presumably §4): the weak-convergence statement is given for fixed r, but the dependence of the limiting measure ν_{p,α} on the choice of coordinate system (first r coordinates) is not discussed. A brief note that the result is rotationally invariant only after suitable re-labeling would clarify the scope.
  4. [Asymptotics] Asymptotic formulas for V_{j(n)}(B_p^n) as n→∞: the error term or rate of convergence is not indicated. Adding a remark on whether the approximation is uniform in α or only pointwise would strengthen the high-dimensional claims.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, the assessment of significance, and the recommendation of minor revision. The report does not list any specific major comments under the MAJOR COMMENTS section. Consequently we have no point-by-point rebuttals to offer. We remain ready to incorporate any minor editorial or typographical changes the editor may request.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives explicit integral formulas for the intrinsic volumes V_j(B_p^n) in terms of the special function F_p from an explicit characterization of the curvature measures Φ_j of coordinate-weighted ℓ_p-balls together with formulas for their mixed moments. This characterization is presented as the main tool and is used to obtain both the volume formulas and the Maxwell-Poincaré-Borel limit theorem via weak convergence of the first r coordinates under the normalized curvature measure, with the limiting measure ν_{p,α} obtained directly from those moments. The derivations reduce correctly to independent known formulas for the special cases p=2 (ellipsoids), crosspolytopes, and boxes. No self-definitional steps, fitted inputs renamed as predictions, load-bearing self-citations, imported uniqueness theorems, smuggled ansatzes, or renamings of known results are present. The chain relies on standard properties of convex bodies and curvature measures without reducing the target results to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivations rely on standard facts from convex geometry (intrinsic volumes, curvature measures, mixed moments) and analysis (integral representations, weak convergence). No free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

axioms (1)
  • domain assumption Standard properties of intrinsic volumes and curvature measures for convex bodies hold and admit explicit mixed-moment formulas for coordinate-weighted ℓ_p-balls.
    Invoked as the main tool underlying the explicit formulas and limit theorems.

pith-pipeline@v0.9.0 · 5771 in / 1300 out tokens · 66674 ms · 2026-05-13T04:50:03.411205+00:00 · methodology

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