Estimates for the Gelfand widths of intersections of finite-dimensional balls

A.A. Vasil'eva

REVIEW 3 major objections 4 minor 18 references

Estimates for the Gelfand widths of intersections of finite-dimensional balls

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2411.14799 v2 pith:KAICTGS2 submitted 2024-11-22 math.FA

A.A. Vasil'eva This is my paper · ORCID

classification math.FA MSC 41A4646B20

keywords GelfandwidthsKolmogorovlinearintersectionsofballsfinite-dimensionalspacesorderestimatesSobolevclassesJohndomains

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 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 $11$, 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

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

This 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.

Recommended: send to peer review, with a referee who knows both Gluskin's method and the Kolmogorov/Gelfand duality.

Referee Report

3 major / 4 minor

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.

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 (3)

[§4, Proof of Theorem 3] 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.
[§5, Proof of Theorem 5] 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.
[§4, Proof of Theorem 3, arbitrary family passage] 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.

minor comments (4)

[Introduction and throughout] 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.
[§1, Theorem 2] 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.
[§4, Proof of Theorem 2] 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.
[§1, Theorem 3] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction; the central estimates are derived from duality, Hölder inclusions, and Gluskin's method, with some general cases delegated to the author's earlier Kolmogorov-width results as proof tools.

full rationale

The main estimates (Theorems 1, 2, and 4) are proved in detail in §4. The λ parameters in (3) and (4) are fixed by the interpolation relation 1/q=(1−λ)/p_i+λ/p_j, not fitted to the target widths, so the results are not self-defined. Upper bounds come from Theorem B (single-ball Gelfand/linear width estimates) and Theorem C (Hölder inclusion), while lower bounds come from the Gluskin-type Lemma 2 and Theorem A duality; these are external or explicitly proved tools. The only in-scope concern is that the most general statements, Theorem 3 (arbitrary family) and Theorem 5 (Sobolev classes), delegate parts of their lower-bound arguments to the author's earlier Kolmogorov-width papers: 'The lower estimate for Gelfand widths of the intersection of the finite family of balls can be proved similarly as in [18, Proposition 1]; here we apply Theorems 1 and 2. For the intersection of an arbitrary number of balls, we argue as in [17, §5]' and 'We argue similarly as in [18], replacing the Kolmogorov widths by the Gelfand widths.' This is self-citation and an omitted-proof/correctness risk, since the dual object for Gelfand widths is a sum of balls rather than an intersection and the transfer is not shown in the paper. However, it is not circular: [17] and [18] concern Kolmogorov widths and are not obtained by assuming the desired Gelfand estimate. A failed transfer would make those proofs incomplete, not make the theorem equivalent to its input. No fitted parameter is renamed as a prediction, and no known result is merely relabeled. Score 1 reflects the minor self-citation/load-bearing proof delegation without any definitional circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to any target value; all order constants depend only on the fixed exponents and domain parameters. The lambda parameters are functions of the given p_i and p_j through the interpolation equation, not free fits. The axioms are standard width-theory results plus an unproved transfer assumption from the author's prior Kolmogorov-width work. No new entities such as forces, particles, or conserved quantities are introduced.

assumptions (5)

standard math Duality of Gelfand and Kolmogorov widths: d_n(B_X, Y) = d_n(B_{Y*}, X*) (Theorem A from Ioffe and Tikhomirov).
Used in Lemma 2 to convert the Gelfand width of the intersection into a Kolmogorov width in the dual space, which is then estimated from below.
standard math Known single-ball width estimates for l_p balls (Theorem B from Gluskin, Pietsch, Stesin).
Used repeatedly for upper bounds and for reductions to the single-ball case, including the p >= q, 2 <= p <= q, and 1 < p <= 2 <= q regimes.
standard math Galeev-Holder inclusion: ν1 B_{p1} ∩ ν2 B_{p2} ⊂ ν1^{1-λ} ν2^λ B_q when 1/q = (1-λ)/p1 + λ/p2 (Theorem C).
This inclusion supplies the upper bounds in Theorems 2, 3, and 4 by embedding an intersection of two balls into a single B_q ball.
standard math Malykhin-Ryutin product-of-octahedra estimate: d_n(B^{m,k}_{2,∞}, l^{m,k}_{∞,1}) ≍ k (Theorem D).
This external result is the core of the lower-bound proof in Theorem 2 for the case where exponents straddle q; it is quoted without proof.
ad hoc to paper The reduction and discretization arguments of [17, Section 5] and [18, Proposition 1] carry over from Kolmogorov widths to Gelfand widths.
The proofs of Theorem 3 and Theorem 5 assert that the earlier arguments apply 'similarly' to Gelfand widths, but the transfer is not demonstrated in this paper.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Estimates for the Gelfand widths of intersections of finite-dimensional balls." pith.science (2026). https://pith.science/paper/KAICTGS2

@misc{pith2026241114799,
  author       = {Pith},
  title        = {Pith review of: Estimates for the Gelfand widths of intersections of finite-dimensional balls},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KAICTGS2}},
  note         = {Machine review of arXiv:2411.14799}
}

read the original abstract

In this paper, we obtain order estimates for the Gelfand widths of intersections of finite-dimensional balls under some conditions on parameters.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 12 canonical work pages

[18]

Kolmogorov widths of an intersection of a ﬁnite f amily of Sobolev classes

A. A. Vasil’eva, “Kolmogorov widths of an intersection of a ﬁnite f amily of Sobolev classes”, Izv. Math. , 88:1 (2024), 18–42. 18

work page 2024
[17]

Kolmogorov widths of an intersection of a family o f balls in a mixed norm

A. A. Vasil’eva, “Kolmogorov widths of an intersection of a family o f balls in a mixed norm”, J. Appr. Theory , 301 (2024), article 106046

work page 2024
[1]

The Kolmogorov diameter of the intersection of clas ses of peri- odic functions and of ﬁnite-dimensional sets

E.M. Galeev, “The Kolmogorov diameter of the intersection of clas ses of peri- odic functions and of ﬁnite-dimensional sets”, Math. Notes , 29:5 (1981), 382– 388

1981
[2]

Kolmogorov widths of classes of periodic functions o f one and several variables

E.M. Galeev, “Kolmogorov widths of classes of periodic functions o f one and several variables”, Math. USSR-Izv. , 36:2 (1991), 435–448

work page 1991
[3]

On widths of a Euclidean ball

A.Yu. Garnaev and E.D. Gluskin, “On widths of a Euclidean ball”, Dokl.Akad. Nauk SSSR , bf 277:5 (1984), 1048–1052 [Sov. Math. Dokl. 30 (1984), 200–20 4] 17

work page 1984
[4]

On some ﬁnite-dimensional problems of the theory o f diame- ters

E.D. Gluskin, “On some ﬁnite-dimensional problems of the theory o f diame- ters”, Vestn. Leningr. Univ. , 13:3 (1981), 5–10 (in Russian)

work page 1981
[5]

Norms of random matrices and diameters of ﬁnite-d imensional sets

E.D. Gluskin, “Norms of random matrices and diameters of ﬁnite-d imensional sets”, Math. USSR-Sb. , 48:1 (1984), 173–182

work page 1984
[6]

Duality of convex functions and extr emum prob- lems

A.D. Ioﬀe, V.M. Tikhomirov, “Duality of convex functions and extr emum prob- lems”, Russian Math. Surveys , 23:6 (1968), 53–124

work page 1968

Show all 18 references

[7]

The diameters of octahedra

B.S. Kashin, “The diameters of octahedra”, Usp. Mat. Nauk 30:4 (1975), 251– 252 (in Russian)

1975
[8]

The widths of certain ﬁnite-dimensional sets and cla sses of smooth functions

B.S. Kashin, “The widths of certain ﬁnite-dimensional sets and cla sses of smooth functions”, Math. USSR-Izv. , 11:2 (1977), 317–333

1977
[9]

On some properties of matrices of bounded operat ors from the space ln 2 into lm 2

B.S. Kashin, “On some properties of matrices of bounded operat ors from the space ln 2 into lm 2 ”, Izv. Akad. Nauk Arm. SSR, Mat. 15 (1980), 379–394 (in Russian)

1980
[10]

A formula of Gaus s in the theory of the method of least squares

A.N. Kolmogorov, A.A. Petrov, Yu.M. Smirnov, “A formula of Gaus s in the theory of the method of least squares”, Izvestiya Akad. Nauk SSSR. Ser. Mat. 11 (1947), 561–566 (in Russian)

1947
[11]

The Product of Octahedra is Badly A pproxi- mated in the l2, 1-Metric

Yu.V. Malykhin, K.S. Ryutin, “The Product of Octahedra is Badly A pproxi- mated in the l2, 1-Metric”, Math. Notes , 101:1 (2017), 94–99

2017
[12]

Widths and rigidity of unconditional s ets and random vectors

Yu.V. Malykhin, K.S. Ryutin, “Widths and rigidity of unconditional s ets and random vectors”, Izvestiya: Mathematics , 89:2 (2025) (to appear)

2025
[13]

s-numbers of operators in Banach space

A. Pietsch, “ s-numbers of operators in Banach space”, Studia Math., 51 (1974), 201–223

1974
[14]

On the best approximations of given classes of f unctions by ar- bitrary polynomials

S.B. Stechkin, “On the best approximations of given classes of f unctions by ar- bitrary polynomials”, Uspekhi Mat. Nauk , 9:1(59) (1954) 133–134 (in Russian)

1954
[15]

Aleksandrov diameters of ﬁnite-dimensional sets and of classes of smooth functions

M.I. Stesin, “Aleksandrov diameters of ﬁnite-dimensional sets and of classes of smooth functions”, Dokl. Akad. Nauk SSSR , 220:6 (1975), 1278–1281 [Soviet Math. Dokl.]

1975
[16]

Kolmogorov widths of intersections of ﬁnite-dim ensional balls

A. A. Vasil’eva, “Kolmogorov widths of intersections of ﬁnite-dim ensional balls”, J. Compl. , 72 (2022), article 101649

2022

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [18]

Kolmogorov widths of an intersection of a ﬁnite f amily of Sobolev classes

A. A. Vasil’eva, “Kolmogorov widths of an intersection of a ﬁnite f amily of Sobolev classes”, Izv. Math. , 88:1 (2024), 18–42. 18

work page 2024

[2] [17]

Kolmogorov widths of an intersection of a family o f balls in a mixed norm

A. A. Vasil’eva, “Kolmogorov widths of an intersection of a family o f balls in a mixed norm”, J. Appr. Theory , 301 (2024), article 106046

work page 2024

[3] [1]

The Kolmogorov diameter of the intersection of clas ses of peri- odic functions and of ﬁnite-dimensional sets

E.M. Galeev, “The Kolmogorov diameter of the intersection of clas ses of peri- odic functions and of ﬁnite-dimensional sets”, Math. Notes , 29:5 (1981), 382– 388

1981

[4] [2]

Kolmogorov widths of classes of periodic functions o f one and several variables

E.M. Galeev, “Kolmogorov widths of classes of periodic functions o f one and several variables”, Math. USSR-Izv. , 36:2 (1991), 435–448

work page 1991

[5] [3]

On widths of a Euclidean ball

A.Yu. Garnaev and E.D. Gluskin, “On widths of a Euclidean ball”, Dokl.Akad. Nauk SSSR , bf 277:5 (1984), 1048–1052 [Sov. Math. Dokl. 30 (1984), 200–20 4] 17

work page 1984

[6] [4]

On some ﬁnite-dimensional problems of the theory o f diame- ters

E.D. Gluskin, “On some ﬁnite-dimensional problems of the theory o f diame- ters”, Vestn. Leningr. Univ. , 13:3 (1981), 5–10 (in Russian)

work page 1981

[7] [5]

Norms of random matrices and diameters of ﬁnite-d imensional sets

E.D. Gluskin, “Norms of random matrices and diameters of ﬁnite-d imensional sets”, Math. USSR-Sb. , 48:1 (1984), 173–182

work page 1984

[8] [6]

Duality of convex functions and extr emum prob- lems

A.D. Ioﬀe, V.M. Tikhomirov, “Duality of convex functions and extr emum prob- lems”, Russian Math. Surveys , 23:6 (1968), 53–124

work page 1968