{"id":"8a0f6ab1-c18a-45a5-8b9e-70be81685d7e","arxiv_id":"2411.14799","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Order-sharp Gelfand and linear width estimates are established for intersections of l_p balls in parameter ranges not covered by the earlier literature.","lead":"This paper proves sharp order estimates for Gelfand widths of intersections of finite-dimensional balls in several new parameter ranges. It also extends the estimates to linear widths and applies them to Sobolev classes on John domains.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The headline formula in Theorem 3 depends on a lower-bound argument for arbitrary families that is not proved in the paper; it is delegated to Kolmogorov-width papers, and the dual object for Gelfand widths is a sum of balls, not an intersection.","rationale":"The reader's weakest_assumption identifies exactly the same transfer concern, and I agree with it. The detailed proofs of Theorems 1, 2, and 4 give real support for the finite special cases, but they do not imply the arbitrary-family statement with constants depending only on p̂. The deferred lower bound is not a cosmetic omission: in Gelfand widths, duality replaces the intersection by the dual sum-ball, so the 'similarly' citations carry a genuine correctness risk. No additional internal inconsistency was found in the detailed arguments; the leftover comment in the proof of Theorem 1 is minor relative to this delegation. The appropriate disposition is therefore to keep the reader's CONDITIONAL verdict: the paper should either supply the missing lower-bound derivation for Theorem 3 or locate an explicit Gelfand-width version of the cited arguments with the duality step written out.","tokens_in":15745,"tokens_out":25550,"duration_ms":236318,"concrete_test":"Reconstruct the lower-bound proof of Theorem 3(1) for a finite family with exponents straddling 2, e.g. r=3, p1=1.5, p2=2.5, p3=4, q=3, and weights satisfying (2), using only Lemma 2 and the reductions of §4. In particular, apply Theorem A to pass to the dual sum-ball ∑ν_α^{-1}B_{p'_α} and check whether the two-level vector construction from Lemma 2, with the appropriate supporting functional, yields the claimed minimum over α; verify the constants do not degrade with r or with gaps between p̂ and the other exponents. If a step requires the dual object to be an intersection rather than a sum, the cited Kolmogorov-width argument does not cover the Gelfand case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The order estimate in Theorem 3 is the central claim. The upper bound is routine: it follows from single-ball estimates (Theorem B) and the Hölder inclusion (Theorem C). The lower bound is the load-bearing part, and it is not proved in the manuscript. For a finite family the paper says the lower estimate 'can be proved similarly as in [18, Proposition 1]', and for an arbitrary family 'we argue as in [17, §5]'. Both citations are about Kolmogorov widths. Under Theorem A, the Gelfand width d_n(∩_α ν_α B_{p_α}^N, l_q^N) equals the Kolmogorov width of the dual unit ball, whose unit ball is the closed convex hull/sum ∑_α ν_α^{-1} B_{p'_α}^N, not an intersection of balls. The reduction and discretization in [17, §5] are written for intersections, and it is not automatic that they survive this duality: the extremal two-level vectors, the norm comparisons, and the uniformity of constants in p̂ = inf_α p_α all have to be re-verified for the sum-ball. This matters because Theorem 3 is the paper's most general finite-dimensional statement and Theorem 5 inherits the same style of delegation. If the transfer fails, the general statements of Theorems 3 and 5 are unsupported even though Theorems 1, 2, and 4 are proved in detail.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Gelfand n-widths of intersections of scaled finite-dimensional l_p balls in l_q^N, for n at most a constant fraction of N. The main finite-dimensional results are Theorems 1 and 2, giving sharp order estimates for finite families under monotonicity conditions on the radii and exponents, and Theorem 3, which claims the same type of estimate for arbitrary families, together with linear-width corollaries. Theorem 4 gives sharp two-ball estimates in l_2^N in a range of radius ratios, and Theorem 5 transfers the finite-dimensional estimates to Sobolev classes on John domains. The proofs of Theorems 1, 2, and 4 are presented in detail, using a generalized Gluskin method developed in Lemma 2 and Proposition 1, together with the Malykhin--Ryutin product-of-octahedra result. The proofs of Theorem 3 and Theorem 5, however, delegate the lower-bound arguments to the author's earlier Kolmogorov-width papers.","tokens_in":16022,"tokens_out":7010,"duration_ms":65784,"significance":"If the main estimates are correct, the paper gives order-sharp Gelfand widths with constants depending only on the infimum of the exponents, and it extends the known Kolmogorov-width theory to Gelfand widths in nontrivial parameter ranges. The detailed proofs of Theorems 1, 2, and 4, and the reusable Lemma 2 and Proposition 1, are genuine strengths. The central risk is the transfer of lower-bound discretization and reduction arguments from Kolmogorov widths to Gelfand widths through duality: the dual object of an intersection of balls is a convex hull of balls, not an intersection, and the cited arguments in [17] and [18] are written for intersections. Because this transfer is exactly what supports the two most general statements, Theorems 3 and 5, the manuscript needs additional work before the advertised scope is fully established.","major_comments":[{"comment":"The lower-bound part of Theorem 3 is asserted rather than proved. The proof says that for a finite family the lower estimate 'can be proved similarly as in [18, Proposition 1]' and that for an arbitrary family 'we argue as in [17, §5]'. Both cited results are for Kolmogorov widths. Under Theorem A, d_n(∩_α ν_α B_{p_α}^N, l_q^N) = d_n(B_{q'}^N, X^*), where the unit ball of X^* is the closed convex hull of ∪_α ν_α^{-1} B_{p'_α}^N, i.e. a sum-type body rather than an intersection of balls. The reduction to extremal two-level vectors, the norm comparisons, and the uniformity of the constants in p̂ = inf_α p_α all have to be re-verified for this convex-hull dual body. Since Theorem 3 is the paper's most general finite-dimensional statement, this delegation is load-bearing, not a mere presentation issue.","section":"§4, Proof of Theorem 3"},{"comment":"The proof of Theorem 5 states 'We argue similarly as in [18], replacing the Kolmogorov widths by the Gelfand widths.' Because [18] is a Kolmogorov-width paper and the same intersection-to-hull duality issue arises after applying Theorem A, the Sobolev-class lower bounds in Theorem 5 are unsupported unless the transfer is written out. In particular, assertions 3(b), 4, and 5 depend both on the finite-dimensional model estimates of Theorems 1, 2, and 4 and on the discretization/reduction arguments of [18]; neither is supplied in Gelfand-width form.","section":"§5, Proof of Theorem 5"},{"comment":"Even if the finite-family lower bound is granted, the passage from a finite family to an arbitrary family A in Theorem 3 is only justified by 'we argue as in [17, §5]'. That argument is again a Kolmogorov-width argument, and for an arbitrary family the infimum in the formula may not be attained, so a limiting or exhausting argument is needed. The manuscript does not indicate how the duality with the convex-hull body behaves under such a passage, nor whether the constants depending only on p̂ survive. This compounds the gap identified above and should be addressed explicitly.","section":"§4, Proof of Theorem 3, arbitrary family passage"}],"minor_comments":[{"comment":"Several inequality symbols are corrupted in the text, for example 'p /greaterorequalslantq' and 'ν1 /greaterorequalslantν2'; the manuscript should be typeset so that all mathematical symbols render correctly.","section":"Introduction and throughout"},{"comment":"Theorem 2 states 2 ≤ q ≤ ∞ together with 2 < p_1 < ... < p_r and p_1 < q < p_r; these hypotheses imply q > 2, so the statement should say 2 < q ≤ ∞ to avoid a vacuous edge case.","section":"§1, Theorem 2"},{"comment":"In the proof of Theorem 2 the authors say it suffices to prove the lower estimate for N = 2^m and n ≤ N/2, while the theorem is stated for n ≤ N/4; a short explanation of the scaling that reconciles these ranges would improve readability.","section":"§4, Proof of Theorem 2"},{"comment":"In Theorem 3.1 the additional condition for linear widths, namely 1/q + 1/p_α ≤ 1 for all α, is stated only in prose; it would be clearer to include it in the display or immediately after the formula.","section":"§1, Theorem 3"}],"recommendation":"major_revision","confidential_remarks":"The main issue is technical completeness: the central general statements, Theorem 3 and Theorem 5, delegate their lower bounds to previous Kolmogorov-width papers without verifying that the reduction survives the duality between intersections of balls and convex hulls of balls. I do not see evidence of a circular or fitted argument; the λ parameters are determined by interpolation equations, not by fitting. If the author can supply the missing Gelfand-width proofs, the results are likely sound and appropriate for publication. The detailed treatment of Theorems 1, 2, and 4 suggests the missing transfer is probably repairable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The useful part of this paper is the finite-dimensional work in Theorems 1, 2, and 4, where the proofs are written out and the new parameter ranges (p1 < 2 ≤ q and 2 < p1 < q < pr) are genuinely new. Lemma 2 is a nice generalization of Gluskin's averaging method, and the explicit computation of the supporting functional for the max-p norm is clean. Theorem 4 for two balls in l2 with p1 < 2 < p2 is a real addition. The Sobolev-class application in Theorem 5 gives plausible rates on John domains.\n\nThe soft spot is exactly where the reader put it: Theorem 3, the advertised general result, does not prove its lower bound. It says the finite-family case follows 'similarly' from [18, Prop. 1] and the arbitrary-family case 'as in [17, §5]', and Theorem 5 delegates the entire argument to [18]. Both cited arguments are for Kolmogorov widths of intersections. The Gelfand width of an intersection dualizes to a Kolmogorov width of a sum of balls under the norm whose unit ball is the convex hull. The reduction to two-level vectors, the norm comparisons, and the uniformity in p-hat all live on the intersection side in those papers; they do not transfer to the sum-ball automatically. That is real work, and the paper does not supply it.\n\nThis makes the paper conditional rather than bad. Theorems 1, 2, and 4 are solid and self-contained (modulo Theorem B, which is standard). The paper is also honest: no fitted parameters, no circularity, and the references to the author's earlier work are legitimate for the Kolmogorov side. I would not cite Theorem 3 in its current form, but I would cite Theorems 1, 2, and 4. The right referee outcome is major revision: ask for a complete proof of the Theorem 3 lower bound, either in the paper or as an appendix that re-verifies the reduction for the dual sum-ball, and similarly for Theorem 5.\n\nRecommended: send to peer review, with a referee who knows both Gluskin's method and the Kolmogorov/Gelfand duality.","headline":"Solid new finite-dimensional results in Theorems 1, 2, and 4, but the general Theorem 3 lower bound is delegated to Kolmogorov-width arguments that do not automatically survive the duality to sums of balls.","tokens_in":16606,"tokens_out":3494,"would_cite":true,"duration_ms":30299,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["41A46","46B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves sharp order estimates for the Gelfand n-width of intersections of finite-dimensional balls, reducing the width, up to constants depending on the smallest exponent, to the minimum of single-ball widths after an…","keywords":["Gelfand widths","Kolmogorov widths","linear widths","intersections of balls","finite-dimensional spaces","order estimates","Sobolev classes","John domains"],"falsifier":"For $q=2$, $p_1=3/2$, $p_2=4$, $\\nu_1=1$, $\\nu_2=N^{-1/6}$, and $n=1$, Theorem 4(1) predicts $d_1(\\nu_1 B_{3/2}^N\\cap \\nu_2 B_4^N,\\ell_2^N)\\asymp N^{-1/15}$ (the other candidate, $N^{1/3}$, is larger). Computing this width to high precision for a range of $N$ (e.g., $N=4,16,64,256$) would settle the claim: the values should track $N^{-1/15}$ up to a constant independent of $N$; any clearly different power law refutes it.","tokens_in":15465,"feed_emoji":"📐","tokens_out":18667,"duration_ms":153540,"temperature":0.7,"pith_summary":"This paper aims to determine, up to constants that may depend only on the smallest exponent, the Gelfand $n$-width of an intersection $\\cap_{\\alpha\\in A}\\nu_\\alpha B_{p_\\alpha}^N$ in $\\ell_q^N$, for $n\\le N/2$ (or $n\\le N/4$ in one case). It proves that when $2\\le q\\le\\infty$ and $1<p_\\alpha\\le q$ for all $\\alpha$ with $\\inf_\\alpha p_\\alpha>1$, the width is, up to constants, $\\inf_\\alpha \\nu_\\alpha\\min\\{1,n^{-1/2}N^{1/p'_\\alpha}\\}$. When the exponents are all at least $2$ and lie on both sides of $q$, the width and the linear width are, up to constants, the minimum of three terms, including geometric means $\\nu_i^{1-\\lambda}\\nu_j^\\lambda$; a separate result handles two balls in $\\ell_2^N$ with $1<p_1<2<p_2$. These finite-dimensional orders are then transferred to Gelfand widths of intersections of Sobolev classes on John domains, yielding the same power-law rates as the corresponding Kolmogorov widths.","feed_headline":"Gelfand width of ball intersections is a single minimum","feed_subtitle":"The width equals the smallest single-ball width after an inverse-square-root cutoff, up to a constant.","key_machinery":"The lower bounds are carried by a generalization of the averaging method of [4]: Lemma 2 chooses a vector $\\hat x$ with $s$ equal nonzero entries, identifies its supporting functional in the dual of $X=(\\mathbb{R}^N,\\max_j \\nu_j^{-1}\\|\\cdot\\|_{\\ell_{p_j}^N})$, and averages squared distances over the group $G=S_N\\times\\{-1,1\\}^N$ acting by permutations and sign changes. Lemma 1 supplies the quadratic convexity inequality $\\|x+h\\|^2\\ge \\|x\\|^2/2+2\\|x\\|f_x(h)+c\\|h\\|^2$ that makes the averaging effective; the final lower bound is the infimum over $t\\ge0$ of a quadratic whose linear coefficient involves $n^{1/2}s^{1/2-1/q}N^{-1/2}\\|I\\|_{X^*\\to\\ell_2^N}$. Upper bounds use the inclusion putting the intersection inside a single ball with geometric-mean radius (Theorem C) plus known single-ball width orders (Theorem B); Theorem D, the product-of-octahedra estimate of [11], supplies the lower bound in the exponent-straddling case. Theorem 5 transfers the finite-dimensional orders to Sobolev classes by the same reduction used in the earlier Kolmogorov-width treatments.","core_discovery":"The central claim is a reduction principle. In the parameter range $2\\le q\\le\\infty$, $1<p_\\alpha\\le q$, $\\inf p_\\alpha>1$, Theorem 3(1) states that for $n\\le N/2$ $$d_n\\bigl(\\cap_{\\$\\alpha$\\in A}\\nu_\\$\\alpha$ B_{p_\\$\\alpha$}^N,\\ell_q^N\\bigr)\\asymp_{\\hat p}\\inf_{\\$\\alpha$\\in A}\\bigl(\\nu_\\$\\alpha$\\min\\{1,$n^{{-1/2}}$$N^{{1/p'_\\alpha}}$\\}\\bigr),$$ where $\\hat p=\\inf_\\alpha p_\\alpha$. Theorem 3(2), for $p_\\alpha\\ge2$ with exponents on both sides of $q$, gives the same order for Gelfand and linear widths: $$\\min\\Bigl\\{\\inf_{p_\\$\\alpha$\\ge q}\\nu_\\$\\alpha$ $N^{{1/q-1/p_\\alpha}}$,\\ \\inf_{p_\\$\\alpha$\\le q}\\nu_\\$\\alpha$,\\ \\inf_{p_\\$\\alpha$<q<p_\\$\\beta$}\\nu_\\$alpha^{{1-\\lambda_{\\alpha\\beta}}$}\\nu_\\$beta^{{\\lambda_{\\alpha\\beta}}$}\\Bigr\\},$$ with $1/q=(1-\\lambda_{\\alpha\\beta})/p_\\alpha+\\lambda_{\\alpha\\beta}/p_\\beta$. Theorems 1, 2, and 4 establish the same type of sharp order for finite families and for the two-ball case in $\\ell_2^N$. Theorem 5 uses these estimates to obtain power-law orders for Gelfand widths of intersections of Sobolev classes on John domains.","pith_inferences":["If the transfer arguments used for Theorems 3 and 5 are valid, the reduction principle should extend to other symmetric finite-dimensional bodies, such as intersections of Orlicz balls or mixed-norm balls, whenever the dual norm and the identity-operator norm can be controlled; the $n^{-1/2}$ factor is the signature of the Euclidean averaging step.","A natural continuation is to determine the exact transition for two balls in $\\ell_2^N$ in the intermediate ratio range $N^{1/p_1-1/2}<\\nu_1/\\nu_2\\le N^{1/p_1-1/p_2}$, where Theorem 4 currently covers only $n\\le a_0(\\nu_1/\\nu_2)^{2\\lambda-2}N$; one would expect the width to interpolate between the geometric-mean value and the clipped single-ball value.","The fact that constants depend on $\\hat p=\\inf p_\\alpha$ rather than on each exponent suggests a stability phenomenon: rates remain unchanged as long as exponents stay bounded away from $1$, which is useful for problems with a continuum of smoothness parameters.","The Gelfand/Kolmogorov duality in Theorem A means the same machinery could be turned around to estimate the Kolmogorov widths of dual intersections, providing a check of the orders by computing the dual set."],"forward_implications":["For $2\\le q\\le\\infty$ and $p_\\alpha\\le q$ with $\\inf p_\\alpha>1$, the width of the whole intersection is, up to a constant depending on $\\hat p$, the minimum of the single-ball widths after the $n^{-1/2}$ clipping; additional constraints cannot make the order worse.","In the exponent-straddling case $p_\\alpha\\ge2$, Gelfand and linear widths have the same order, and the optimum is either a single ball on one side of $q$ or a geometric-mean interpolation between one ball below $q$ and one above $q$.","For two balls in $\\ell_2^N$ with $1<p_1<2<p_2$, the width is $\\min\\{\\nu_1^{1-\\lambda}\\nu_2^\\lambda,\\nu_1 n^{-1/2}N^{1/p_1'}\\}$ in the small-ratio range, and $\\nu_1^{1-\\lambda}\\nu_2^\\lambda$ in the large-ratio range for sufficiently small $n$.","For intersections of Sobolev classes on John domains, the Gelfand widths have the same power-law orders as the Kolmogorov widths: e.g., $n^{-r_1/d}$ when all $p_j\\ge q$, and $n^{-\\min\\{\\theta_1,\\theta_2\\}}$ in the transition cases.","Under the extra duality condition $1/q+1/p_\\alpha\\le1$, the same order estimates hold for linear widths, so linear and Gelfand widths coincide in order in these regimes."],"supporting_citations":[{"why":"Supplies the averaging method generalized in Lemma 2 to produce lower bounds for Gelfand widths.","marker":"[4]"},{"why":"Provides the single-ball Gelfand and linear width orders used in Theorem B for upper and lower estimates.","marker":"[5]"},{"why":"Earlier result on Kolmogorov widths of intersections; also supplies the inclusion used for upper bounds (Theorem C).","marker":"[1]"},{"why":"Duality formula (Theorem A) relating Gelfand and Kolmogorov widths, used throughout the proofs.","marker":"[6]"},{"why":"Product-of-octahedra estimate (Theorem D) used for lower bounds when exponents straddle $q$.","marker":"[11]"},{"why":"Earlier Kolmogorov-width estimates for intersections of balls that this paper extends to Gelfand widths.","marker":"[16]"},{"why":"Its Section 5 is invoked to pass from finite families to arbitrary families of balls in Theorem 3.","marker":"[17]"},{"why":"Its Proposition 1 and reduction arguments are invoked for the finite-family lower bounds and for the Sobolev-class application.","marker":"[18]"}],"fun_headline_variants":["Intersection widths reduce to the tightest ball's cutoff","Gelfand width of intersections: the bottleneck ball wins","Sharp order for Gelfand widths of ball intersections","Cutoff reduction yields exact order for Gelfand widths","Ball intersection widths determined by the weakest ball"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"In the proofs of Theorems 3 and 5, the lower estimates are asserted to follow by 'arguing as in [18]' (or '[17, Section 5]') with Kolmogorov widths replaced by Gelfand widths, without reproducing the reduction and discretization arguments; if those arguments do not transfer to Gelfand widths, the general statements of Theorems 3 and 5 are unsupported.","fun_headline_variants_meta":{"raw":{"variants":["Intersection widths reduce to the tightest ball's cutoff","Gelfand width of intersections: the bottleneck ball wins","Sharp order for Gelfand widths of ball intersections","Cutoff reduction yields exact order for Gelfand widths","Ball intersection widths determined by the weakest ball"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000616,"raw_usage":{"total_tokens":2827,"prompt_tokens":876,"completion_tokens":1951,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":492,"completion_tokens_details":{"reasoning_tokens":1874}},"tokens_in":492,"tokens_out":1951,"duration_ms":15747,"temperature":1.0,"reasoning_tokens":1874,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:52:35.453714+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $q=2$, $p_1=3/2$, $p_2=4$, $\\nu_1=1$, $\\nu_2=N^{-1/6}$, and $n=1$, Theorem 4(1) predicts $d_1(\\nu_1 B_{3/2}^N\\cap \\nu_2 B_4^N,\\ell_2^N)\\asymp N^{-1/15}$ (the other candidate, $N^{1/3}$, is larger). Computing this width to high precision for a range of $N$ (e.g., $N=4,16,64,256$) would settle the claim: the values should track $N^{-1/15}$ up to a constant independent of $N$; any clearly different power law refutes it.","supporting_citations":[{"cited_title":"On some finite-dimensional problems of the theory o f diame- ters","cited_arxiv_id":null,"evidence_quote":"Supplies the averaging method generalized in Lemma 2 to produce lower bounds for Gelfand widths."},{"cited_title":"Norms of random matrices and diameters of finite-d imensional sets","cited_arxiv_id":null,"evidence_quote":"Provides the single-ball Gelfand and linear width orders used in Theorem B for upper and lower estimates."},{"cited_title":"Duality of convex functions and extr emum prob- lems","cited_arxiv_id":null,"evidence_quote":"Duality formula (Theorem A) relating Gelfand and Kolmogorov widths, used throughout the proofs."},{"cited_title":"The Product of Octahedra is Badly A pproxi- mated in the l2, 1-Metric","cited_arxiv_id":null,"evidence_quote":"Product-of-octahedra estimate (Theorem D) used for lower bounds when exponents straddle $q$."},{"cited_title":"Kolmogorov widths of intersections of finite-dim ensional balls","cited_arxiv_id":null,"evidence_quote":"Earlier Kolmogorov-width estimates for intersections of balls that this paper extends to Gelfand widths."},{"cited_title":"Kolmogorov widths of an intersection of a family o f balls in a mixed norm","cited_arxiv_id":null,"evidence_quote":"Its Section 5 is invoked to pass from finite families to arbitrary families of balls in Theorem 3."},{"cited_title":"Kolmogorov widths of an intersection of a finite f amily of Sobolev classes","cited_arxiv_id":null,"evidence_quote":"Its Proposition 1 and reduction arguments are invoked for the finite-family lower bounds and for the Sobolev-class application."}],"review_version":1}