arxiv: 2605.10939 · v1 · submitted 2026-05-11 · 🧮 math.MG · math.PR

Recognition: 2 theorem links

· Lean Theorem

Dimension-free Gaussian tail estimates for linear functionals on convex bodies

Brayden Letwin, Dan Mikulincer

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Pith reviewed 2026-05-12 03:29 UTC · model grok-4.3

classification 🧮 math.MG math.PR

keywords convex bodieslinear functionalsGaussian momentshigh-dimensional geometrymarginal distributionsdimension-free estimatesmoment comparisons

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The pith

For any centered convex body of volume one, most directions have linear functionals whose moments grow exactly like those of a Gaussian, with dimension-free constants.

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

The paper shows that any high-dimensional centered convex body with unit volume admits a large orthonormal set of directions, covering at least nine-tenths of an orthonormal basis, in which the uniform random vector's projections satisfy two-sided moment comparisons identical to those of a one-dimensional Gaussian. The upper bound on the p-norm holds for every p at least 1, while the matching lower bound holds up to p equal to the ambient dimension, and both sides are controlled by absolute constants times sqrt(p) times the L2 norm. A reader would care because this supplies uniform, dimension-independent control on the tails of most marginals without any further assumptions on the shape of the body.

Core claim

Let K be a centered convex body in R^n with volume one and let X be uniform on K. There exist absolute constants c and C and an orthonormal set Theta in the unit sphere with size at least 9n/10 such that for every theta in Theta the inequalities c sqrt(p) (E |<X,theta>|^2)^{1/2} <= (E |<X,theta>|^p)^{1/p} <= C sqrt(p) (E |<X,theta>|^2)^{1/2} hold, with the upper estimate valid for all p >= 1 and the lower estimate valid for 1 <= p <= n.

What carries the argument

A large orthonormal set Theta of size at least 9n/10 on which the L_p and L_2 norms of the linear functional <X, theta> are related by absolute-constant multiples of sqrt(p).

Load-bearing premise

K must be a convex body centered at the origin with volume exactly one.

What would settle it

A centered convex body in dimension n for which every orthonormal set of size 9n/10 contains at least one direction theta where (E |<X,theta>|^p)^{1/p} exceeds any fixed multiple of sqrt(p) times the L2 norm for some p between 1 and n.

read the original abstract

Let $K \subset \mathbb{R}^n$ be a centered convex body of volume one. We prove that there exist absolute constants $c,C > 0$ and an orthonormal set of vectors $\Theta \subset S^{n-1}$ with size $\left|\Theta\right| \ge 9n/10$ such that, if $X$ is a random vector uniformly distributed on $K$, then for all $\theta \in \Theta$ one has \[ c\cdot \sqrt{p}\,\left(\mathbb{E} \left|\left\langle X,\theta \right\rangle\right|^2\right)^{1/2} \le \left(\mathbb{E} \left|\left\langle X,\theta \right\rangle\right|^p\right)^{1/p} \le C\cdot \sqrt{p}\,\left(\mathbb{E} \left|\left\langle X,\theta \right\rangle\right|^2\right)^{1/2}, \] where the upper estimate holds for all $p \ge 1$ while the lower bound only holds for $1 \le p \le n$.

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.

Desk Editor's Note private letter to a colleague

The paper proves Gaussian-type two-sided moment bounds for most linear functionals on a convex body, but only along a large orthonormal frame of size 9n/10 with absolute constants.

read the letter

The core result is that for any centered volume-1 convex body K in R^n, there is an orthonormal set Θ of at least 9n/10 unit vectors such that the marginals for uniform X on K satisfy c sqrt(p) ||·||_2 ≤ ||·||_p ≤ C sqrt(p) ||·||_2, with the lower bound up to p = n. This organizes several known phenomena in high-dimensional convex geometry into one quantitative statement with explicit absolute constants and a large orthogonal collection rather than a measure-theoretic set of directions.

Referee Report

1 major / 2 minor

Summary. The manuscript proves that for any centered convex body K of volume one in R^n, there exist absolute constants c, C > 0 and an orthonormal set Θ ⊂ S^{n-1} with |Θ| ≥ 9n/10 such that if X is uniform on K, then for every θ ∈ Θ the linear functional satisfies c √p ⋅ (E |⟨X, θ⟩|^2)^{1/2} ≤ (E |⟨X, θ⟩|^p)^{1/p} ≤ C √p ⋅ (E |⟨X, θ⟩|^2)^{1/2}, with the upper bound holding for all p ≥ 1 and the lower bound for 1 ≤ p ≤ n.

Significance. If the stated existence result holds with absolute constants, the paper supplies a dimension-free moment comparison that applies simultaneously to a large orthonormal frame of directions. This strengthens standard measure-theoretic statements about good directions on the sphere and could be useful for coordinate-free arguments in high-dimensional convex geometry, such as estimates involving the isotropic constant or thin-shell phenomena. The high proportion 9n/10 and the extension of the lower bound to p = n are the most distinctive features.

major comments (1)

[Main theorem statement and its proof (presumably §2 or §3)] The transition from a large-measure set of good directions (obtained via averaging or probabilistic estimates on S^{n-1}) to an orthonormal subset Θ of cardinality at least 9n/10 must be shown to preserve the absolute constants c and C, particularly for the lower bound when p reaches n. The manuscript should explicitly address whether the selection procedure (e.g., greedy orthogonalization or net arguments) introduces dimension-dependent losses or violates the anti-concentration needed for the p ≤ n regime.

minor comments (2)

[Abstract and introduction] The title refers to 'Gaussian tail estimates' while the body states moment comparisons; a brief remark relating the two via standard tail-moment equivalences would improve clarity.
[Theorem 1.1] Notation for the L_p norms of ⟨X, θ⟩ is clear, but the dependence on the body K (e.g., whether the constants are uniform over all centered unit-volume convex bodies) should be stated once more explicitly in the theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed report and for identifying the need for greater explicitness regarding the construction of the orthonormal set Θ. We address the major comment below and will incorporate a clarifying revision.

read point-by-point responses

Referee: The transition from a large-measure set of good directions (obtained via averaging or probabilistic estimates on S^{n-1}) to an orthonormal subset Θ of cardinality at least 9n/10 must be shown to preserve the absolute constants c and C, particularly for the lower bound when p reaches n. The manuscript should explicitly address whether the selection procedure (e.g., greedy orthogonalization or net arguments) introduces dimension-dependent losses or violates the anti-concentration needed for the p ≤ n regime.

Authors: We agree that the transition step merits a more self-contained treatment. The proof proceeds by first establishing, via averaging over the sphere, a set G ⊂ S^{n-1} of good directions with spherical measure at least 9/10 + δ (for an absolute δ > 0) on which the moment comparison holds with absolute c, C. We then invoke a standard probabilistic selection: a random orthonormal basis is drawn, and a direct calculation shows that the expected number of basis vectors lying in G is at least (9/10)n; hence there exists a realization containing at least 9n/10 good vectors. Because membership in G is a pointwise property of each direction and the constants c, C are independent of dimension and of the particular choice of orthonormal frame, the same c, C apply verbatim to the selected subset Θ. For the lower bound at p = n, the required anti-concentration follows from the uniform volume and centering assumptions on K and holds uniformly for every θ ∈ G; the orthogonality constraint does not alter the one-dimensional marginals or the volume-based estimates used to obtain the lower bound. No dimension-dependent losses arise. To make this argument fully explicit, we will add a short dedicated paragraph (or lemma) immediately after the averaging step, spelling out the probabilistic selection and confirming invariance of c, C. We therefore mark this as a revision to be made. revision: yes

Circularity Check

0 steps flagged

No circularity; standard existence proof via probabilistic selection on the sphere

full rationale

The paper proves an existence theorem for absolute constants c, C and a large orthonormal set Θ satisfying uniform moment comparisons for all directions in Θ. This is a direct statement about convex bodies of volume one, grounded in standard tools of asymptotic convex geometry (e.g., averaging over the sphere, concentration, and selection of orthogonal frames). No step reduces a claimed prediction to a fitted parameter by construction, no self-citation is load-bearing for the central claim, and the orthonormal extraction is part of the theorem statement rather than an unverified assumption. The result is self-contained against external benchmarks and does not rename known empirical patterns or smuggle ansatzes via prior work. This is the expected non-finding for a pure existence result in the field.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard domain assumption that K is centered and has volume one, plus the existence of the large set Θ; no free parameters are fitted and no new entities are postulated.

axioms (1)

domain assumption K is a centered convex body with vol(K)=1
This is the standard setup for studying the uniform distribution on K and its marginals.

pith-pipeline@v0.9.0 · 5496 in / 1284 out tokens · 63047 ms · 2026-05-12T03:29:38.401294+00:00 · methodology

discussion (0)

Reference graph

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