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Gaussian Width of Convex Sets via Integral Decompositions, Projections, and the Distribution of Intrinsic Volumes

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Pith's one-line read The Gaussian width of a convex set — the expected maximum of a Gaussian process over the set — is determined, up to universal constants, by the diameter and the peak index of its intrinsic volumes, bypassing generic chaining.

desk verdict Real new decomposition identities and an attractive peak-index characterization, but the main theorem's proof leans on an unsourced Poisson-log-concavity step that needs a citation or proof. read the letter →

arxiv 2603.02714 v2 pith:UMBNTC7Q submitted 2026-03-03 math.PR cs.ITmath.ITmath.STstat.TH

Reese Pathak , Nikita Zhivotovskiy This is my paper · ORCID
classification math.PRcs.ITmath.ITmath.STstat.TH MSC 60G1552A2060D05
keywords GaussianwidthintrinsicvolumesWillsfunctionalconvexbodiesmetricprojectionssequencemodelDudleyentropyintegralminimaxestimation
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

This paper tries to establish that the Gaussian width of a convex set — the expected maximum of a Gaussian process indexed by the set — is governed by a single geometric feature: the mode, or peak index, of its intrinsic volumes. The authors produce two exact decompositions of the width, one through fixed points of a penalized local width and one through metric projections of a Gaussian vector on rescaled copies of the set, and show neither requires the optimal partition constructions of generic chaining. Combining these identities with the Wills functional and a recent comparison between its logarithm and the logarithm of a single intrinsic volume, they conclude that w(T) is within universal multiplicative constants of i*·diam(T), where i* is the index where the intrinsic volumes of T/diam(T) peak. If correct, the result gives a parameter-free, purely geometric formula for a quantity central to high-dimensional statistics, probability, and signal processing.

What carries the argument

The load-bearing objects are two exact decompositions. In the first, the width is the value at r(σ), the unique maximizer of the strongly concave map r ↦ w(ξ; T∩rB₂) − r²/(2σ), plus (1/2)∫_σ^∞ r(ν)²/ν² dν; in the second, it is a penalized supremum sup_{t∈T}{⟨t,ξ⟩ − ‖t‖²/(2σ)} plus an integral of expected squared projection norms ‖Π_{T/ν}(ξ)‖². The link from these identities to geometry is the Wills functional W(T) = Σᵢ Vᵢ(T): a variational representation bounds the penalized supremum by σ log W(T/(σ√(2π))), and a recent inequality shows log W is within a constant factor of log maxᵢ Vᵢ exactly when σ < w(T). The peak intrinsic index — the mode of the intrinsic-volume distribution — is the nam

What would settle it

Compute w(T)/(i*·diam(T)) for convex bodies whose intrinsic volumes are known in closed form — crosspolytopes, products of simplices, random polytopes — and check whether the ratio stays within fixed universal constants as the dimension grows; a single family where the ratio escapes that band would refute Theorem 2.3.

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Extended reading notes

Core claim

The authors set out to prove that the Gaussian width of a compact convex set is determined, up to universal constants, by its intrinsic volumes: for any σ below the width, w(T) ≍ σ·i*_σ plus an integrated tail term, where i*_σ is the peak index of the sequence V_i(T/(σ√(2π))), and at the scale σ = diam(T)/√(2π) this becomes w(T) ≍ i*·diam(T), with i* the mode of the intrinsic volumes of T/diam(T). The width is first split by two exact identities — one through fixed points of a penalized local width and one through squared norms of metric projections of a standard Gaussian onto rescaled copies of T — and the proof shows these decompositions agree up to constants in the Gaussian case. The firs

Load-bearing premise

The characterization imports, as external facts, a bound on the penalized Gaussian supremum by a constant times the localized critical radius squared and a factor-8 comparison between the log-Wills functional and the log of its largest intrinsic volume (valid when w(T) ≥ 2σ), together with Poisson log-concavity of the intrinsic-volume sequence; if any of these fails in the regime σ < w(T), the conclusion w(T) ≍ i*·diam(T) can collapse.

Editorial extensions

If this is right

  • Up to universal constants, computing the Gaussian width reduces to locating the mode of the intrinsic-volume sequence of T/diam(T); no optimal admissible partitions or chaining constructions are needed.
  • At every scale σ below the width, w(T) is approximately σ times the peak index at that scale plus an integrated tail term, giving a scale-by-scale geometric picture of the width.
  • For a centrally symmetric body whose minimum-volume enclosing ellipsoid is the Euclidean ball, the peak index must be at least a constant multiple of √(log e d) and at most of order √d; the ℓ₁ ball and the Euclidean ball attain opposite extremes.
  • Local and global forms of the Dudley entropy integral, and of Sudakov minoration, agree up to universal constants for arbitrary bounded sets, not just convex ones.
  • The classical Dudley entropy integral is loose exactly when the integrated risk of the least-squares estimator at the origin is far below the integrated minimax risk over the set.

Reading between the lines

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

  • The projection-based decomposition is proved for an arbitrary random vector, so the same integral scheme may extend to non-Gaussian and even non-convex index sets, where projections replace intrinsic volumes as the geometric carrier.
  • An immediate conjecture beyond the paper: any two convex bodies whose intrinsic-volume sequences share the same peak index and the same first intrinsic volume have Gaussian widths of the same order, since the full profile beyond the mode appears irrelevant.
  • The result reframes 'effective dimension' of a convex body as the peak index of V_i(T/diam T), offering a practical route to width estimates from tables of intrinsic volumes for standard bodies (balls, crosspolytopes, products of simplices) without any chaining computation.
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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

2 major / 3 minor

Summary. The paper develops exact decompositions of the Gaussian width of a compact convex set T ⊂ R^d containing the origin. Theorem 2.1 expresses w_ξ(T) through a family of fixed points r(σ) of a quadratically penalized localized width, and Theorem 2.2 gives a parallel decomposition in terms of metric projections onto rescaled copies of T. Both are derived from convex-analysis envelope theorems and the fundamental theorem of calculus. In the Gaussian case the paper compares the two decompositions and then links the first term to the Wills functional and intrinsic volumes, using work of Vitale and Mourtada. The central result, Theorem 2.3, asserts that w(T) is, up to universal constants, equal to σ times the peak intrinsic-volume index i*_σ plus an integrated local-width/projection term, and in particular w(T) ≍ i* · diam(T), where i* is the maximizer of V_i(T/diam(T)). The paper also develops connections to statistical minimax rates, local and global Dudley/Sudakov bounds, and information-theoretic proofs of Sudakov minoration, and gives a detailed treatment of the crosspolytope.

Significance. If the central characterization is correct, Theorem 2.3 is a striking geometric description of Gaussian width that bypasses generic chaining: the width of a convex body is determined, up to constants, by the diameter and the mode of its intrinsic-volume sequence. The exact decomposition identities in Theorems 2.1 and 2.2 are elegant, appear to be correctly derived from first principles, and are likely to be useful independently. The paper is also careful to use absolute, derived constants rather than fitted parameters, and it transparently identifies which external inequalities are load-bearing. The crosspolytope and ellipsoid analyses are informative and the information-theoretic perspective on Sudakov minoration is suggestive. However, the peak-index theorem rests on an intrinsic-volume inequality that is asserted without proof or reference, and a key lemma in the local/global comparison section is false as stated; these issues prevent the manuscript from being accepted in its current form.

major comments (2)
  1. [§4.2, Eqs. (31) and (33)] The proof of Theorem 2.3 uses 'Poisson-log-concavity of the intrinsic volume sequence' to assert V_i(T) ≤ V_1(T)^i / i! and V_i(T)/V_{i-1}(T) ≤ V_1(T)/i, but this property is neither proved nor given a supporting reference. Equation (31) is used in the upper bound to control log V_{i*_σ}; equation (33) is used to conclude i*_σ ≤ V_1(T/(σ√2π)) = w(T)/σ, which supplies the lower bound σ i*_σ ≤ w(T). Without these inequalities the peak-index characterization loses both its upper-bound control and its lower-bound justification. The property may be a standard ultra-log-concavity result, but the manuscript must provide a proof or a precise citation. As written, this is an unverified load-bearing assertion.
  2. [§5.6.3, Lemma 5.4] The proof of the upper bound I>_τ(δ) ≤ min{1/τ, √h(δ)/δ} uses the implication 'if f(δ)>ν then ε(ν)≤δ'. This requires the set {ε : f(ε)≤ν} to be downward closed, i.e., monotonicity of f(ε)=ε√h(ε). No such monotonicity is stated, and it is false for valid local packing entropies: for T=[0,1], h_loc jumps from log 3 to log 2 at ε=1/2; taking δ just below 1/2 and ν=0.43, one has ε(ν)>δ while ν<δ√h(δ). Thus Lemma 5.4 is false as stated. Proposition 5.2, Corollary 5.2, and the claims that the local Dudley/Sudakov forms are equivalent and that the Dudley bound is recovered all depend on this lemma. The proof needs a different argument or a corrected statement.
minor comments (3)
  1. [§4.2, proof of Theorem 2.3] The proof is written for the integral term involving r(ν) and states that the projection-integral version is handled 'in exactly the same way'. Since Theorem 2.3 states both equivalences, please add the short justification using (8) and Proposition 3.3 so the second equivalence is explicit.
  2. [§5.2, Remark 5.3] Remark 5.3 refers to a 'short note that the current authors recently announced' as [47] for a claimed simplified proof of the lower bound in the majorizing measures theorem. This is not verifiable from the manuscript; either include the argument or rephrase the remark as a conjecture/future work.
  3. [§3.1, Proposition 3.1] The proof invokes [43, Proposition 4.1] and then [43, Lemma 8.2] to adapt it to centrally symmetric convex bodies. It would help the reader to state the exact conditions of the cited lemma, since the central-symmetry assumption is used at this step.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor self-citation in the crosspolytope projection lemma; the central width/peak-index derivation is independent.

  1. self citation load bearing [Section 6.3.3, Lemma 6.4]
    "The following claim follows easily from convex duality; we omit the proof, but note that it uses the same argument as in Lemma 1 in [4]."

    Lemma 6.4 supplies the explicit soft-thresholding characterization of the projection onto the l1 ball that is then used in Lemma 6.11 and Proposition 6.1 to bound E||Pi_{B_1^d}(sigma xi)||^2 and to illustrate Theorem 2.2 on the crosspolytope. The proof is omitted and justified only by a citation to the authors' own work [4] (R. Pathak is a co-author). However, this self-citation is confined to an illustrative example; the paper's central results (Theorems 2.1, 2.2, and 2.3) are derived from exact identities and from external cited theorems (Chatterjee, Vitale, Mourtada, McMullen), not from [4]. Thus this is a minor self-citation that is load-bearing only for the peripheral crosspolytope projection computation.

full rationale

The main derivation chain is self-contained. Theorem 2.1 is an exact identity obtained by applying the envelope theorem to the variational definition of r(sigma), and Theorem 2.2 follows pointwise from differentiating h_sigma(x) = sup_{t in T}(<x,t> - ||t||^2/(2sigma)), with no fitted parameters or hidden normalization. Theorem 2.3 combines these identities with external results: Vitale's representation (28), McMullen's inequality (29), and Mourtada's log-Wills inequalities (32)/(17). The peak-index characterization is not definitional: i*_sigma is the mode of the intrinsic-volume sequence, whereas the width enters via V1, and the inequalities V_i <= V1^i/i! and V_i/V_{i-1} <= V1/i are invoked as Poisson-log-concavity properties rather than derived from the theorem being proved. The paper does state eqs. (31) and (33) without proof or citation, which is an unsupported assertion and a correctness/verification risk, but it is not circularity: these are external facts about intrinsic volumes, not the paper's own outputs. The only self-citation that functions as an actual proof substitute is Lemma 6.4, which is used in the crosspolytope example; since the central theorems do not depend on it, the appropriate score is 2.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No free parameters are fitted; all constants are absolute and derived. The central claims rest on several recent external theorems, especially from Mourtada and Chatterjee, which are used as black boxes with precise references. No new particles, forces, or invented quantities are introduced.

assumptions (8)
  • standard math Steiner formula and intrinsic volumes; V_1(T)=sqrt(2pi) w(T)
    Used in Section 2.1 to define intrinsic volumes and connect V_1 to Gaussian width.
  • domain assumption Unimodality and Poisson log-concavity of intrinsic volumes: V_i <= V_1^i/i! and V_i/V_{i-1} <= V_1/i
    Recent results cited from [5,34,43]; used to locate the peak intrinsic index in Theorem 2.3.
  • domain assumption Mourtada [43, Prop 2.1]: sigma log W(T/(sigma sqrt(2pi))) <= 8 sigma log max_i V_i when w(T) >= 2 sigma
    Load-bearing in the upper bound of Theorem 2.3, eq. (32).
  • domain assumption Mourtada [43, Prop 4.1]: penalized Gaussian supremum over T/sigma is at most 140 (r*(sigma)/sigma)^2
    Used to prove first-term equivalence in Gaussian case, eq. (17).
  • domain assumption Chatterjee [19] / Prasadan-Neykov [50, Lemma 2.1]: r(sigma)^2 is comparable to max{sigma^2, E||Pi_T(sigma g)||^2}
    Used to compare integrated terms in the two decompositions, eq. (27) and Proposition 3.3.
  • standard math Talagrand's majorizing measures theorem: w(T) is comparable to gamma_2(T)
    Background characterization and motivation, eq. (1).
  • standard math McMullen's inequality and Vitale's Wills-functional representation
    Used in eqs. (28)-(29) to connect the first decomposition term to the Wills functional and to w(T).
  • standard math Fano, Le Cam, and Bretagnolle-Huber information-theoretic inequalities
    Used in Sudakov minorations and the metric-entropy characterization in Section 5.2 and Appendix A.

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Pith. "Pith review of Gaussian Width of Convex Sets via Integral Decompositions, Projections, and the Distribution of Intrinsic Volumes." pith.science (2026). https://pith.science/paper/UMBNTC7Q

@misc{pith2026260302714,
  author       = {Pith},
  title        = {Pith review of: Gaussian Width of Convex Sets via Integral Decompositions, Projections, and the Distribution of Intrinsic Volumes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UMBNTC7Q}},
  note         = {Machine review of arXiv:2603.02714}
}
abstract

We revisit the problem of bounding the expected supremum of a canonical Gaussian process indexed by a convex set $T \subset \mathbf{R}^d$. We develop two decompositions for the Gaussian width, based on the geometry of the index set. The first decomposition involves metric projections of Gaussians onto rescaled copies of $T$. The second involves fixed points arising from a quadratically penalized variant of the local width. Neither decomposition directly invokes generic chaining constructions. Our results make use of recent work in geometric analysis and Gaussian processes. The work of Chatterjee [Ann. Statist., 2014] characterizes the behavior of the metric projection of a Gaussian random vector onto rescaled copies of $T$ with a variational problem involving localized Gaussian widths. We use these bounds to develop decompositions of the Gaussian width using the local metric structure of $T$. Second, we leverage the work of Vitale [Ann. Probab., 1996] to form a connection between the Wills functional (and hence the intrinsic volumes of $T$) and the first terms that appear in our decompositions. Finally, invoking recent work by Mourtada [J. Eur. Math. Soc., 2025] on the logarithm of the Wills functional, we show that the width is controlled by a single, ''peak index'' of the intrinsic volumes. In the worst case, our bound recovers a local form of the classical Dudley integral.

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    The estimator pθpYq“ arg min ηPMε }η´ Y}2 2 enjoys the following risk bound sup θ‹PT EY„Npθ‹,σ2Idq}pθpYq´ θ‹}2 2À ε2. Note that Proposition A.1 does not require T to be convex; however if T is not convex, then the supremum over radii δě ε in the definition of the local entropy...

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