DIAMETERS OF SOME FINITE-DIMENSIONAL CLASSES OF SMOOTH FUNCTIONS

Transcription

1 Izv. Akad. Nauk SSSR Math. USSR Izvestija Ser. Mat. Tom 41 (1977), No. 2 Vol. 11 (1977), No. 2 DIAMETERS OF SOME FINITE-DIMENSIONAL CLASSES OF SMOOTH FUNCTIONS UDC B. S. KASIN SETS AND Abstract. Estimates of the diameters of certain sets in the Banach spaces L^iO, 1) and /" are given; in particular, the orders of the diameters d n (W r, ), Ρ < q, r > 1, are completely determined. Bibliography: 18 titles. Introduction and formulation of the basic theorems^) Let X be a Banach space and Κ a compact centrally symmetric subset of X. The quantity ά η (Κ, X) = inf sup inf \x y\\, where the inf runs over all subspaces L n of X having dimension < n, is called the Kolmogorov η-diameter of the set Κ in X. Furthermore, l n p denotes the space R", equipped with the norm * η By Bp we denote the unit ball in /", and by W r (r > 1, 1 < ρ < ) we denote the well-known class of /--smooth functions defined on the segment [0, 1] (when r is an integer, it consists of the functions whose derivatives of order r \ are absolutely continuous and for which for the definition of the class W r when r is not an integer, see [18]). AMS (MOS) subject classifications (1970). Primary 52A05, 46E30; Secondary 46E15, 52A4S, 52A40. (!) Some of the theorems in this paper were announced previously in [7]. Copyright 1978, American Mathematical Society

2 318 Β. S. KASIN THEOREM 1. Let 1 < η < m <. THEOREM 2. Let 1 < ρ < q < q > 2 and ψ > 1. 77ien 4,(1^,^(0, THEOREM 3. Let 1 < ρ <q <. 77ien Γ «/ P>2, "^, *7 P<2. 1, P \ «/ P>2. The assertion of Theorem 3 for ρ = 1 or q < 2 is not new: it follows at once from the estimates (2) and (3'). Besides these theorems, which have to do with intersections, we prove the following result: THEOREM 4. For any η > 1 there exists in the space R" an orthogonal transformation Τ such that \χ In, *2 χ ΕΞ R n. Theorem 2 finishes the solution of the problem of determining the orders of the quantities d n (W r, L q ) and, in combination with the already known results, implies that for r> 1 rr r, if q or 2 < ρ < q, d n {W r p,l Q )x 1 1 a p c (1) TV if The first results on the diameters of classes of smooth functions were obtained by Kolmogorov [8] (p = q = 2). Steckin [9] obtained for an estimate of the diameters of W[ in L 2 and W x in L the equality («< m) (2) In 1960 Tihomirov calculated the exact values of the diameters d n (W^, C), and then Tihomirov, Babadzanov and Makovoz (see [5] and [10] -[12]) proved the inequalities (1) in the case p~> q. For 1 <p < q < 2 the relations (1) were obtained by Ismagilov [13] ; ( 2 ) The symbols C, C', and Β in the following denote various absolute positive constants.

3 DIAMETERS OF CLASSES OF FUNCTIONS 319 he also observed that the equivalence d n {W r, i,? )X(i'' + 1 ' r 1 '' fails for ρ = 1, q =. p Before the appearance of the present paper the asymptotic behavior of d n {W r, p Lq ), ρ <q, q > 2, was known only for ρ = 1, r > 2 (Gluskin [14]). In [14] it was shown that for an exact estimate of d n {W\, C) it is sufficient to get a good estimate of the diameter d n {B, I ). Later, Maiorov [15] carried out this reduction of the problem of determining the order of the quantity d n (Wp, L q ) to the corresponding "finite-dimensional" problem for all ρ and q(p<q). The "finite-dimensional" problem of estimating the diameters d n {B ', Ι ), ρ < q, also has independent interest. A sufficiently accurate estimate for d n {B^, I ) was known only for 1 < ρ < q < 2 and for 1 = ρ < q <. In the first case it follows directly from (2), and in the second case it is a consequence of the following result of the author (see [16]): d n (B?, O<-%, ηι λ^η^ιη, λ>0. (3) We mention that for application to an estimate of the diameters d n (W[, C) for r > 2 it is even sufficient to use the earlier estimate of Ismagilov [13] : O-n ipi, too) <= η For a proof of Theorem 2 we use the following obvious corollary of Theorem 1: COROLLARY 1. For m> η and 1 < ρ < 2 ^ ί \ (4) For application of Corollary 1 the power of the factor (1 + \n(m/n)) appearing in (4) is not of importance to us, and we shall not concern ourselves with a determination of the exact value of this power; we mention only (see [17], and also the estimate (3)) that for P= 1 d n (BT,O< PROOF OF THEOREM 1. It suffices for us to prove Theorem 1 only in the case when -Η In η where the constant C is arbitrarily large, since if (5) does not hold, then the theorem follows from the obvious estimate d n (B, I ) < 1. Let A' = {a jj }" =l JL l be a matrix with η rows and m columns (n < m). We denote by e f (1 < i < w) the columns of the matrix A'. An important point in the proof of the theorem is the construction of a matrix A' having the following two properties: *) Any η columns e i,..., e t οϊα' are linearly independent. **) For any set e,,..., e ( - (1 < i k < m) the coefficients in the expansion

4 320 Β. S. KASIN η satisfy the inequality (λ = {λ 1;..., λ η }) In We prove Theorem 1 under the assumption that a matrix A' satisfying *) and **) has been constructed. For χ Ε R m and 1 <i<mwe let (x) t denote the rth coordinate of the vector x. We consider the η-dimensional subspace L C R m spanned by the row vectors ί_ν, }" of A', and we show that for any point ζ Ε Β there is an element y G L such that (7) We make use of the following well-known corollary of Helly's theorem on the intersection of convex sets (for a proof see [2], 1): if y\,..., y' n and ζ are vectors in R m, m> n, then for the distance in the metric of F from ζ to the subspace generated by y\,..., y' n to be less than or equal to p 0 it is necessary and sufficient that for any set i v..., i n+1, 1 < i k < m, there is a linear combination Σ"^^ such that <Po. We now choose an arbitrary set of columns {e ( - } ί J of A'. Moreover, let η We choose a nonsingular matrix {b r A" =l S(^M fc = f?' such that r=h \ 0< r - *<») (10) (this is possible, since, by the property *) of the matrix A', detio^. } φ 0). We now define in the space L a new basis {y' r }", setting By (io), We determine the values of the quantities (y' t ) i. Using (9) and (10), we have

5 DIAMETERS OF CLASSES OF FUNCTIONS 321 = Σ =Σ b '< Σ λ * Κ Consequently, in the m χ w matrix Λ that determines the basis {y' r }" the following («+ 1) χ η matrix is cut out by the columns with the numbers i v..., i n+ χ : the first η columns of it form an identity matrix, and the (n + l)th column is the column λ = Let ζ B%. For 1 < r < η we set We estimate the quantities (z - Σ?^),-, 1 < k < η + 1. Using (9) and (6) for 1 < k < «, we have (11) I λ IU «Η» \\ χ 1η Μ *1 *1 For Λ = «+ 1 it is easy to verify that 2 = 0. (11') ίη Since the set of columns {e t }%=\ was chosen arbitrarily, the estimate (7) follows from (11), (11 ), and the above corollary of Helly's theorem. Thus, Theorem 1 follows from the existence of a matrix A' satisfying the conditions *) and **). For the construction of such a matrix A' we shall need several auxiliary statements. LEMMA 1. For any integer η and any a > 0 it is possible to find a set of vectors Ω η (α) = {ζ { }Ί with z t e S",( 3 ) 1 < ζ < k, such that k<(c ογ 1 )" and for any y e S" there is a number i for which Without regard to the size of the constant C, Lemma 1 is easy to prove directly; to save space we refer to [6], where the question of the size of C is considered. Suppose that we are given integers q and m (1 < q < m) and a number a > 0. In ( 3 ) By S n we denote the unit sphere in fl.

6 322 Β. S. KASIN R m we define a system of vectors l m (q, a) in the following way. There exist C^ ^-dimensional subspaces L of R m that are defined as follows: On the unit euclidean sphere of each such subspace L we define for a given number a a system of vectors Ω (α) satisfying the condition of Lemma 1. The union of all the vectors of these systems gives the set Sl m (q, a) It is clear that the number of elements in ^OT (i7, a) is not greater than C^ (C a~ x ) q. have LEMMA 2. For any bilinear form A(x, y) = V" j=i a ij x i y j (x = {x t };y = {y f }) we sup llji!u=w,n=i «2 'a Α (*,ίθ = μ <2. sup A(x,y), x.yea n (i/t) where the set Ω,,(1/6) is determined by the number a = 1/6 in Lemma 1 (therefore C"). Indeed, let ^41 = >1(JC 0, y 0 ), Λ: 0, y 0 & S". Using the property of the system of vectors Ω η (1/6) (see Lemma 1), we find vectors y ε Ω π (1/6) such that \\x -x o \\ n < 1/6 and h Α (χ, y) = A(x o +(x x o ), y o +(y yo))=a (*<>> y o )+A (χ 0, y y 0 ) The lemma is proved. + Α(χ χ 0, y o )+A(x x o, y y o )>A(x o,y o ) -\Α\ '\-\Α\\-\Α\ ±>\Α\.\ LEMMA 3 (see [1], p. 217 and [3], Chapter III, 5, Theorem 8). If P(x) = Z\c k r k (x) is any polynomial in the Rademacher system, then the following assertions are true: 1) There exists an absolute constant C o > 0 such that ^[0,1]:\Ρ(χ)\>Ο 0 (^ c% 2) For any y > 0 l^cl\ J<2. e " r. By \E\ we denote the number of elements in any finite set E, and by N(x), χ = {x t } R", we denote the set of all numbers i, 1 </'<«, such that x t Φ 0. LEMMA 4. //{a, }" is a set of real numbers,

7 DIAMETERS OF CLASSES OF FUNCTIONS 323 / η Υ' Α ( " Σ α Π =»- -ψλύ Κ \/=ι / \<=ι / f/ien for any t > 1 there is a set of integers E t with E t <Z [1, n], \E t \ < η (2ty) 2, such that PROOF. Without loss of generality, we can suppose that We set Then E t = {i: 1 < i< η, α] >η" 1 (2ίγ)" 2 }. and, therefore, Furthermore, Consequently, ί=1 2 α?>1- \>t^al The lemma is proved. We set We introduce the following definitions: let A = {e ij }1L l? x (m > n) be a real matrix. sup / η t m \ z \ 2 SU P 5 1. Σ (12) ιό From the definition it is clear that for m = η we have F(A) = \\A\\. For 1/n < θ < 1 we set Ο(Λ,Θ)= inf ^ 2 '2 ' LEMMA 5. For any numbers a > 0 awd 1 /«< θ < 1 awd matrix A = {e, }^ j y_ j, m 1=1 (13)

8 324 Β. S. KASIN (Α, θ); inf Σ YnF(A)-a. PROOF. Let /=1 4=1 In the set i2 m ([«9], a) we find a vector χ = {x ( } such that JC - JC O m < α and 17V(x-x o ) < [«6»] <«. Then ' 2 Σ m Σ **/ <Σ m 4=1 +Σ /=1 < G μ, e)+k«ί 2 Σ (*'- x ' 9 ) 8 '/) ) The lemma is proved. We make use of the following simple estimate for the number of combinations C^: (14) Indeed, m (m η + 1). m n n\ "" n! By Stirling's formula, n\> n" C ", which proves (14). We proceed to the construction of a matrix A' satisfying the conditions *) and **). Here we use probability arguments. On the set D mn of all m χ η matrices A = {e f/ -}^=1 " =1 with elements equal to ± 1 we introduce a measure that assigns to each matrix A the measure TT m ' n. Then μώ ηιη = 1. For.y >0 let (see (12)) (15) By Lemma 2 and the definition of l m (n, a), f(y)<,)x{a<=d mn : sup () <; C^ C sup μ Μ ez) OT : x.yf=s n [ 2J ' 4,/=l A(x, y)>2-i.y} >2~ 1 -y\. (16) We estimate the right-hand side of (16). Since for x, y G S"

9 DIAMETERS OF CLASSES OF FUNCTIONS 325 it follows that (**/>'= Σ*? ΣriH Let {c k }" easy to see that be the numbers x f. y. (1 < /, / < «), numbered in any order. Then it is μ U e D nn : 2 ε^,-y,- > 2-^1 = μ It e [0, 1 ]: 1 i./=i J I. (17) By part 2) in Lemma 3 and the relation Σ" c\ = 1, the right-hand side of (17) does not exceed 2 e~ y ' 8. Finally, for the function f(y) we obtain the estimate (see (16)) - ^ From (18) it follows directly that for y = C\n + In C^) 1/2 we have f(y) < 1/100, where C' is a sufficiently large absolute constant. Since (see (14)) In C^ < CXn + η \n(m/n)), it follows that (19) Further, for α > 0, l/«< θ < 1 and ζ > 0 we set inf (20) We estimate the magnitude of the function g(0, a, z); for this, we first estimate for any fixed χ G S m the measure 2 x ε «7 1 «By part 1) of Lemma 3 we have for each / (1 < / < n) μ \A<=D mn : c o j > c 0. Consequently, for ζ < 1/2

10 326 Β. S. KASIN μίλεζ?,: S 1 1=1 χ ί ε ί/ >c 0 <C 0 ; / = 1, 2,..., n [zn] (21) From (21) it follows that for any χ S μ 2 xfiu (22) Since i2 m ([w0], α) contains not more than C\ e] {C α 1 ) Μβ elements, (22) implies the following estimate for the function g(d, a, z) (see (20)): g(q. a, C o? (C a-i (23) We simplify the right-hand side of (23). Since lim z^. 0 (Cz l ) z = 1, we find a number z 0 such that for ζ < z r (τ \--C ο where C is the constant from (14). Then, by (14), for ζ < z 0 we have (24) Consequently, for ζ < z 0 (23) can be written as follows: (25) where C = (1 - C o ) 1/2 (1 - C O /2)~ 1/2 < 1 is an absolute constant. We now fix the numbers a and θ, setting α = z 0 C 0 +In ^- -1 (26) (here the constant Β is the same as in (19) and z 0 is defined by (20)) and (27) We show that if the absolute constant B' is sufficiently small, then (see (25)) g(θ, a, C o z)< CL" 9] (Ccr 1 )"" (5/< i (for ζ< z 0 and η >n 0 ). Indeed (see (14), and consider also (5)),

11 DIAMETERS OF CLASSES OF FUNCTIONS 327 exp fin ^ (. ηθ ^ s>b'-n ΛΒ'Ή (-ΘΙηθ)η exp {(in --Inθ) ηθ} <u n, where u <(C)~ 1/3, if β' in (27) is sufficiently small. Further, ηθ where u < (C)" 1 ' 3, if B' in (27) is sufficiently small. Consequently, for the numbers α and θ defined by (26) and (27) and for ζ < z 0 (see (25)) we have g (θ, α, C oz ) < (C)" 8 " (C) 8 " (C) n < (C) s " < 1 (28) for η > n 0. From (19) and (28) it follows that for some constants B, B', 0 < z 0 < 1, 0 < C o < 1, and η > n 0 there exists a matrix Λ = {e ii }^L 1 " =l G Z) mn such that 1) F(i4)<B K» 2) Jnl,.i o - z 0 n, (29) -1 Applying Lemma 5 for this matrix A and the number θ (see also (13)), we get (see (29)) G(A,Q), ini /=1 YnF(A)-a (30) '^j z 0 n. It is easy to see (see (29) and (30)) that by a very small change in the elements of the matrix A = {e^ } it is possible to get an m χ η matrix A' = {e^ } for which the following conditions hold: 1) le.,1 > 1/2, 1 < 1 < m, 1 </<«. 2) F(A') < 2B\/n{\ + ln(m/«)) 1/2, Β an absolute constant; obviously we can assume that Β > 1. 3) G(A', Θ) >Q η, θ = B'(l +ln(w/n))~ 1,2>0 an absolute constant; obviously (31) we can take Q < 1/2. 4) If e i (1 < i < m, e i ε R") is the «th column vector of A', then any set {e i. }^=1 of η columns forms a linearly independent system of vectors in R". **). The matrix A' thus constructed is the desired one, i.e. it satisfies the conditions *) and The condition *) holds by (31), part 4). We prove that the condition **) holds. By

12 328 Β. S. KASIN (5) we can assume here that We choose an arbitrary set {e t }%t J of η + 1 columns of the matrix A'. Since e i ε /?" (1 / < m), there is a linear dependence among these vectors: (32) [by (31), part 4), the coefficient of e t Let fc=i in (33) cannot be zero], (33) We prove the following estimates: -ν. (34) where the constants Q, B, and B' are the same as in (31). From (34) we easily get (6) for the vector λ = {Xj,..., X n } defined by (33). Indeed, if the inequalities (34) have been proved, then i+v +c Since the set {e ( } ί j was chosen arbitrarily, the last estimate implies directly the condition **) for the matrix A. Thus to conclude the proof of Theorem 1 it suffices to prove (34). Since (see (31), part 1)) e (. > y/n/2 (1 < / < m), it follows (see (33), (12), and (31), part 2)) that ' 2 ' v > i.e. which proves (34), a).

13 DIAMETERS OF CLASSES OF FUNCTIONS 329 We now assume that (34), b) does not hold. Then the system of inequalities (35) has a solution t = f 0, where (35') Since for 0 < ν λ <υ 2 we have (see (34), a), and (31), 2)) Μ >=" (' +ta 7) (, + (Χ Using the last inequality and also the inequality (see (31), part 3)) Q < 1/2, we get that If we now apply Lemma 4 for this value ί = t Q > 1 to the set of numbers {X fc }"', then we get that all the numbers {i k }" can be partitioned into two groups Ε and Ε such that: 1) \E\ < η 4y 2 t 2, and consequently (see (35) and (32)) also (36) Σ *ί. (37) (37) From (36) and (37) it follows (see also (13) and (31), part 3)) that ~ Σ - Σ k:i k <=E ^i* η Next (see (12), (34) and (31), part 3)), we have

14 330 Β. S. KASIN ί_ Σ VJI <F(^l') / Σ tiy<f(a')- (- Σ λί Thus we have obtained, beginning from the assumption that (34), b) is false, the result that +1 - Σ but the last estimate contradicts (33). This contradiction proves (34), b), and hence also the whole of Theorem 1. PROOF OF THEOREM 3. We use the following simple inequality: for χ G R 2n and ρ We first give upper bounds for the quantities d n {B 2n, l 2n ). 1) Using Theorem 1 and (38), for 1 < ρ < 2 we have d n (Bf, IT) < d n {Bf, If) < d a (Bf, C) (2«) Γ < C η τ ) Again using Theorem 1 and (38), for 2 < ρ < we have d n (BT, if) < (2nf d n (Bf, Ο < (2nf (2nf ' J d n (Bf, Ο < C η We now establish a lower bound. 1) For ρ > 2 we use the fact that, by (38), the ball B* n contains the set E pq = {x e= R 2n : Ι χ ^ < (2n) J + Γ }. Consequently dn (Bf, IT) > d n (E Pg, if) > (2n) ^ + U n (Bf, If) = (2n)~ ^ + *~. 2) For 1 <p < 2 we use, besides (38), also the equality (2):

15 DIAMETERS OF CLASSES OF FUNCTIONS 331 ά η (BT. IT) > *n (B?, C) >n~^~ ^ dn (ΒΓ. I?) >jfi ^ ^ (? > 2 ) These estimates constitute the assertion of Theorem 3. We mention in addition that for any y > 1 the diameter d n (B l pyn^, / 7 "'), 1 <p < q <, has the same order (for η * ) as d n (B^n, l 7 qn ). PROOF OF THEOREM 2. Using the results of [15], Theorem 2 follows easily from the estimates obtained here for the diameters d n (B, / ). Using the inequality (see [15]) n(w p, V) >C p, r,q Π ir + ) P V d n (B p, l q ) and Theorem 3, we get for q > 2 that (C Pf r, q n- r, if p>2, d a (W r p,v)>\ ( r+ L L) [C p, r, q n l 2 p, if l<p<2. Now we estimate d n (Wl, L q ) from above. We first mention that for ρ > 2 d n (Uv *)< d n (W^,, C) < d n (WJ, C). Therefore it suffices for us to obtain the upper estimates required in Theorem 2 only for 1 <p < 2 and q =. If in Theorem 2 of [15] we substitute the estimate for the diameter d n (B, Ο (1 < ρ < 2) obtained in Corollary 1 of this paper, then for 1 < ρ < 2 we get By the above remark, the last inequality concludes the proof of Theorem 2. Only Theorem 4 remains to be considered. We merely outline a proof of this theorem, since the necessary arguments for the proof have already been used in the proof of Theorem 1. The only new fact, in comparison with Theorem 1, is that in Theorem 4 the averagings are carried out with respect to the group of orthogonal matrices. Let O" be the group of orthogonal matrices of order n, and let μ be Haar measure on this group (see [4]). By mx we denote the usual Lebesgue measure of a set X in the sphere S n (ms" = 1). It is obvious that for any / (1 </ < n) we can choose η j coordinates (x) t of a vector χ = {(x),}" with JC < asjn such that n Ί K*)ij<^p- ο <*<«-/) Using (39) (for example, for; = [n/2]), we can show that for sufficiently small a 0 > 0 and any η we have (39) / (α ΟΙ η) = ηι{χεξs n :\\x\\ n < a 0 γη} < 2Λ ( 4 0 ) Ί Using (40) and the invariance of the measure μ with respect to shifts, we get that for any

16 332 Β. S. KASIN XGS" μ {Α <= 0" : I Ax 1 < a 0 γη) = / (a 0, n) < 2~\ (41) Ί Reasoning just as for the proof of Theorem 1 (see, in particular, Lemma 5), we can get from (41) the existence of a matrix 7G O n such that if \N(x)l < βη, then \\Tx\\ > (*jn, where β and a are positive absolute constants. Then, using Lemma 4, it is not hard to see that such a matrix Τ satisfies the requirements of Theorem 4. We have also the following result, which is close to Theorem 4, but is somewhat simpler: For any positive number θ there exists a constant C e > 0 such that for any η > 1 there is a plane such that if χ & L ne, then «. e C#\ dim.. > η (Ι Θ), l What is more, if we set Z..e = {*e# n :(x), = 0 for t>«(l then for sufficiently small C e > 0 the measure of those re O" for which the plane T(L e ) does not satisfy the last assertion is smaller than 2~ n. The author thanks Professor V. M. Tihomirov for an interesting discussion of these results. Received 27/FEB/76 BIBLIOGRAPHY 1. A. Zygmund, Trigonometric series, 2nd rev. ed., Vol. I, Cambridge Univ. Press, New York, Ludwig Danzer, Branko Griimbaum and Victor Klee, Helly's theorem and its relatives, Proc. Sympos. Pure Math., vol. 7, Amer. Math. Soc, Providence, R. I., 1963, pp V. V. Petrov, Sums of independent random variables, "Nauka", Moscow, 1972; English transl., Springer-Verlag, Berlin and New York, Andre Weil, L'integration dans les groupes topologiques et ses applications, Actualite's Sci. Indust., no. 869, Hermann, Paris, V. M. Tihomirov, Diameters of sets in functional spaces and the theory of best approximations, Uspehi Mat. Nauk 15 (1960), no. 3 (93), ; English transl. in Russian Math. Surveys IS (1960). 6. V. M. Sidel'nikov, New bounds for densest packing of spheres in η-dimensional Euclidean space, Mat. Sb. 95 (137) (1974), ; English transl. in Math. USSR Sb. 24 (1974). 7. B. S. Kasin, The orders of the diameters of some classes of smooth functions, Uspehi Mat. Nauk 32 (1977), no. 1 (193), (Russian) 8. A. Kolmogoroff [A. N. Kolmogorov], Uber die beste Annaherung von Funktionen einer gegebenen Funktionenklasse, Ann. of Math. (2) 37 (1936), S. B. Steckin, On best approximations of given classes of functions by arbitrary polynomials, Uspehi Mat. Nauk 9 (1954), no. 1 (59), (Russian) 10. S. B. Babadzanov and V. M. Tihomirov, Diameters of a function class in an L p -space (p > 1), Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk 11 (1967), no. 2, (Russian) 11. V. M. Tihomirov, Certain problems of approximation theory, Doctoral Dissertation, Moscow State Univ., Moscow, 1969 (Russian); abstract published in Mat. Zametki 9 (1971), ; English transl. in Math. Notes 9 (1971).

17 DIAMETERS OF CLASSES OF FUNCTIONS Ju. I. Makovoz, On a method for estimation from below of diameters of sets in Banach spaces, Mat. Sb. 87 (129) (1972), ; English transl. in Math. USSR Sb. 16 (1972). 13. R. S. Ismagjlov, Diameters of sets in normed linear spaces and the approximation of functions by trigonometric polynomials, Uspehi Mat. Nauk 29 (1974), no. 3 (177), ; English transl. in Russian Math. Surveys 29 (1974). 14. E. D. Gluskin, On a problem concerning diameters, Dokl. Akad. Nauk SSSR 219 (1974), ; English transl. in Soviet Math. Dokl. 15 (1974). 15. V. E. Maiorov, Discretization of the diameter problem, Uspehi Mat. Nauk 30 (1975), no. 6 (186), (Russian) 16. B. S. Kasin, Diameters of octahedra, Uspehi Mat. Nauk 30 (1975), no. 4 (184), (Russian) 17., On Kolmogorov diameters of octahedra, Dokl. Akad. Nauk SSSR 214 (1974), ; English transl. in Soviet Math. Dokl. 15 (1974). 18. S. L. Sobolev, Applications of functional analysis in mathematical physics, Izdat. Leningrad. Gos. Univ., Leningrad, 1950; English transl., Amer. Math. Soc, Providence, R. I., Translated by Η. Η. McFADEN