{"id":"dcb9d31f-1fb3-44be-b75d-7f5cc3573ee4","arxiv_id":"2603.02714","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Gaussian width of a convex set is equivalent, up to universal constants, to its diameter times the index of the largest intrinsic volume at the diameter scale.","lead":"This paper gives exact decompositions of the Gaussian width of a convex set into a local variational term plus an integrated projection term, and proves that the width is essentially controlled by the index of the largest intrinsic volume. The new viewpoint offers bounds that avoid generic chaining and connects the geometry of convex bodies to statistical minimax rates.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central theorem depends on unproved Poisson-log-concavity of intrinsic volumes; if eq. (33) fails, the peak-index lower bound collapses.","rationale":"The reader's conditional verdict rightly flags external inequalities and the Poisson-log-concavity black box. My stress-test concentrates the concern on the Poisson-log-concavity property (eqs. 31 and 33) because it is both unproved and directly used in the lower and upper bounds of the main theorem. The cited Mourtada inequalities are published results and, while load-bearing, are at least externally checkable; the Poisson-log-concavity assertion appears with no proof or reference in the text. The decomposition identities themselves (Theorems 2.1 and 2.2) are proved with envelope theorems and elementary convex analysis, and appear sound. Thus the single most vulnerable point is the intrinsic-volume profile inequality. Since the reader's CONDITIONAL verdict already accounts for exactly this kind of unverified black box, no verdict adjustment is needed: the verdict remains CONDITIONAL. If the concrete test shows the inequality fails, the verdict would need to move to REJECT; if it passes and a citation is found, it could move to ACCEPT. For now, UNCHANGED is the honest verdict.","tokens_in":48237,"tokens_out":36493,"duration_ms":303458,"concrete_test":"Compute exact intrinsic volumes of the standard simplex Δ^d and the crosspolytope B_1^d (using known formulas, e.g., Betke–Henk for B_1^d and standard simplex Steiner-polynomial formulas) for d = 5, 10, 20, 50. For K = T/diam(T), check whether V_i(K)/V_{i-1}(K) ≤ V_1(K)/i for every i = 1,...,d. If any ratio exceeds V_1/i, Theorem 2.3's lower bound is false and the peak-index claim is refuted. If the ratios all satisfy the bound, the remaining concern is the absence of a citation — one should then locate a proof of Poisson-log-concavity of intrinsic volumes for all convex bodies; unless such a proof is found, the theorem is conditional on an unstated fact.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 2.3 is the paper's central claim: Gaussian width is controlled, up to constants, by the peak intrinsic-volume index times the diameter. The proof's lower bound uses the asserted inequality V_i(K)/V_{i-1}(K) ≤ V_1(K)/i, labeled \"Poisson-log-concavity\" (eq. 33), to conclude that the peak index satisfies i*_σ ≤ V_1(K) = w(T)/σ, so σ·i*_σ ≤ w(T). The upper bound uses the companion inequality V_i ≤ V_1^i/i! (eq. 31), and the paper states that Mourtada's inequality (32) also follows from the same Poisson-log-concavity property. Unlike the cited Mourtada results, eqs. (31) and (33) are not sourced or proved in the manuscript. If (33) fails — e.g., for non-centrally-symmetric bodies or in some high-dimensional regime — then the identification of the peak index with w(T)/σ is unjustified, and the diameter specialization w(T) ≍ i*·diam(T) has no lower bound. This is more load-bearing than the external Mourtada inequalities because those are published theorems, whereas (33) is an unverified assertion in the proof itself. The rest of the decomposition machinery (Theorems 2.1 and 2.2) is rigorously derived and is not the source of risk.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":48511,"tokens_out":41582,"duration_ms":340059,"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":[{"comment":"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.","section":"§4.2, Eqs. (31) and (33)"},{"comment":"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.","section":"§5.6.3, Lemma 5.4"}],"minor_comments":[{"comment":"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.","section":"§4.2, proof of Theorem 2.3"},{"comment":"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.","section":"§5.2, Remark 5.3"},{"comment":"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.","section":"§3.1, Proposition 3.1"}],"recommendation":"major_revision","confidential_remarks":"The main geometric decomposition machinery appears sound and the paper is likely to be correct in broad outline, but the proof of Theorem 2.3 currently depends on an unsupported intrinsic-volume inequality, and Lemma 5.4 contains a concrete error. Both are fixable within the manuscript's scope, but they are not merely cosmetic. I would also gently flag that the paper cites several very recent and in one case unpublished works ([47], [67]); the authors should ensure these are available or remove the dependency before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth engaging seriously. The two decompositions (Theorems 2.1 and 2.2) are genuinely new and clean: they turn a one-parameter family of local widths into an exact identity for the global width, and the projection version follows directly from convex analysis. The connection to the Wills functional and the peak index (Theorem 2.3) is refreshing, and the equivalence of local and global Dudley/Sudakov bounds in Section 5 is a useful standalone tool. Constants are explicit, and there is no parameter fitting anywhere.\n\nThe biggest concern, which the stress-test flags correctly, is in the proof of Theorem 2.3. The argument relies on the inequalities V_i(K) ≤ V_1(K)^i/i! and V_i/V_{i-1} ≤ V_1/i, labeled “Poisson-log-concavity,” but these are neither proved nor cited at the point of use. The reader's report missed this. It is a genuine gap: the lower bound in Theorem 2.3 collapses without those inequalities. If they come from the cited paper [5] (which the authors cite earlier for unimodality), they need to say so explicitly. If not, they need a proof. This is fixable, but a referee should demand it.\n\nOther soft spots are minor. Lemma 6.4 is stated without proof, but the reference to [4] provides enough for an expert; a longer proof sketch would help. The novelty boundary with Mourtada's Wills-functional work could be drawn more sharply, but the paper does cite his results where used.\n\nOverall, I think the main theorems are likely correct, and the flaws are not structural. This paper will be useful to people working on Gaussian width bounds, statistical minimax rates, and convex geometry. I would send it to a serious referee and would likely cite it once the Poisson-log-concavity point is resolved.","headline":"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.","tokens_in":48999,"tokens_out":7952,"would_cite":true,"duration_ms":62248,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G15","52A20","60D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Gaussian width","intrinsic volumes","Wills functional","convex bodies","metric projections","Gaussian sequence model","Dudley entropy integral","minimax estimation"],"falsifier":"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.","tokens_in":48092,"feed_emoji":"📐","tokens_out":9534,"duration_ms":83123,"temperature":0.7,"pith_summary":"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.","feed_headline":"Gaussian width reduces to a single peak index of intrinsic volumes","feed_subtitle":"Up to constants, a convex body's Gaussian width equals its diameter times the mode of its intrinsic volumes.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Gaussian width pinned by peak intrinsic volume","Width of convex sets: one integral volume rules all","Peak intrinsic volume dictates Gaussian width","Gaussian width reduced to mode of intrinsic volumes","Convex set width: intrinsic volumes pick the peak"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Gaussian width pinned by peak intrinsic volume","Width of convex sets: one integral volume rules all","Peak intrinsic volume dictates Gaussian width","Gaussian width reduced to mode of intrinsic volumes","Convex set width: intrinsic volumes pick the peak"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000361,"raw_usage":{"total_tokens":1833,"prompt_tokens":839,"completion_tokens":994,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":583,"completion_tokens_details":{"reasoning_tokens":924}},"tokens_in":583,"tokens_out":994,"duration_ms":7207,"temperature":1.0,"reasoning_tokens":924,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T19:17:19.492572+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}