JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS vol. 162, No. 2, 1991 k-β and k-Nearly Uniformly Convex Banach Spaces Denka Kutzarova Different uniform geometrical properties have been defined between the uniform convexity and the reflexivity of Banach spaces. In the present paper we introduce other properties of this type, namely k-β and k-nearly uniform convexity. The definitions, as well as some of the results presented here, are announced in [7]. Sullivan [24] has defined k-uniformly rotund Banach spaces which by a suggestion of Davis are all superreflexive. Fan and Glicksberg [1] have introduced fully k-convex Banach spaces. Let x1,...,xk+1 be vectors in a Banach space X. The k-dimensional volume enclosed by x1,...,xk+1 is given by V (x1, x2,...,xk+1) 1 ··· 1 f (x ) ··· f (x ) 1 1 1 k+1 ∗ = sup . : fi ∈ X , kfik ≤ 1, i =1, 2,...,k . . fk(x1) ··· fk(xk+1) A Banach space X is said to be k-uniformly rotund (k-UR), k ≥ 1, if for each ε > 0 there exists a δ > 0 such that for any xi ∈ X, kxik < 1, i = 1, 2,...,k + 1, k+1 with i=1 xi (k +1) ≥ 1−δ, then V (x1,...,xk+1) <ε. Clearly, 1-UR coincides with uniform convexity. For equivalent definition of k-UR see [11]. P A Banach space X is said to be fully k-convex (kR), k ≥ 2, if for every sequence k {xi} in X such that limn1,...,nk→∞ i=1 xn k = 1, then {xi} is a Cauchy sequence in X. P Another uniform property is the nearly uniform convexity, introduced by Huff [3]. He has proved that the class of nearly uniformly convexifiable spaces is strictly between superreflexive and reflexive Banach spaces. A Banach space X is called nearly uniformly convex (NUC) if for each ε > 0 there is a δ, 0 <δ< 1 such that for any sequence {xn} in the closed unit ball B with sep(xn) = inf{kxn − xmk : n =6 m} > ε, then conv({xn}) ∩ (1 − δ)= ∅. For equivalent definitions of NUC see [19]. 1 The Kuratowski measure of non-compactness α(A)ofaset A in X is the infimum of those ε> 0 for which there is a covering of A by a finite number of sets Ai, such that diam(Ai) <ε. Let X be a Banach space with closed unit ball B. By the drop D(x, B) defined by an element x ∈ X \ B, we mean conv({x} ∪ B) and let R(x, B) = D(x, B) \ B. Rolewicz [18] has proved that X is uniformly convex if and only if for each ε > 0 there is a δ > 0 such that 1 < kxk < 1+δ implies diam(R(x, B)) <ε. In connection with this he has also introduced [19] the property (β). A Banach space X is called to have the property (β) if for each ε> 0 there is a δ > 0 such that 1 < kxk < 1+ δ implies α(R(x, B)) <ε. Rolewicz [19] has shown that UC ⇒ (β) ⇒ NUC. The class of Banach spaces with an equivalent norm with property (β) coincides neither with that of superreflexive spaces (independently done in [4] and [14]) nor with the class of nearly uniformly convexifiable spaces (see [5, 6]). Yu Xin-Tai [25] has proved that k-UR implies NUC. In connection with this result we may ask if k-UR implies (β). The answer is negative. For this purpose we may present first the following. Proposition 1. Let X be the l1-direct sum of Banach spaces Y and Z, where Y is uniformly convex and Z is k-UR. Then X is (k + 1)-UR. This is an addition to the results in [2, 13]. The author was informed by Prof. Bor-Luh Lin that the same result had been also obtained by Yu Xin-Tai. Example 2. There is a 2-UR Banach space which does not possess the property (β). 1 Proof. Let Y = R and Z = l2. Denote by X the l1-direct sum of Y and Z. According to Proposition 1, X is 2-UR. On the other hand, it is easy to observe that X fails to have the property (β) (cf. [19, 14]). If in the above mentioned characterization of uniform convexity by drops [18] we replace diam(R(x, B)) < ε by sup{diam(C) : C ⊂ R(x, B), C convex} < ε, obviously we have again UC. In contrast, if we do a similar substitution in the definition of (β), we obtain: Theorem 3. A Banach space X is NUC if and only if for each ε > 0 there exists a δ > 0 such that 1 < kxk < 1+ δ implies sup{α(C) : C ⊂ R(x, B), C convex} < ε, where B is the closed unit ball of X. Proof. Necessity. Assume the contrary; i.e., there exists an ε > 0, elements xn ∈ X with 1 < kxnk < 1+1/n and convex sets Cn, Cn ⊂ R(xn, B), so that 2 α(Cn) ≥ ε, n = 1, 2,... For every integer n we have that the convex set Cn ∗ is disjoint with B and thus, there is a functional fn ∈ X with kfnk = 1 which separates Cn and B, i.e., f(x) ≥ 1 for every x ∈ Cn. Put En = Cn/(1+1/n). Clearly, En is convex and α(En) ≥ ε/2 for every integer n. By the choice of xn and Cn we obtain easily that f(x) > 1 − 1/n for every x ∈ En, i.e., En ∩ (1 − 1/n)B = ∅, n =1, 2,.... By [19], this means that the norm is not NUC which is a contradiction. Sufficiency. Assume that the norm is not NUC. Then, following [19], we may ∗ find an ε> 0 and a sequence {fn} ⊂ X with kfnk = 1 so that α(S(fn, 1/n)) ≥ ε, n =1, 2,..., ∗ where S(f, δ) = {x ∈ B : f(x) ≥ 1 − δ} for f ∈ X and 0 <δ< 1. Take xn with 1+2/n ≤kxnk ≤ 1+3/n and fn(xn)=1+2/n, n =1, 2,... Then, if we denote by Cn the set xn/2+ S(fn, 1/n)/2, we have that Cn is convex and Cn ⊂ R(xn, B). Moreover, α(Cn) ≥ ε/2 for every n, which is a contradiction. We are now in a position to introduce the notions of k-β and k-nearly uniform convexity. Definition 4. Let k ≥ 1 be an integer. A Banach space X with closed unit ball B is called to be k-β, provided for each ε > 0 there exists a δ > 0 so that 1 < kxk < 1+ δ implies sup{α(C)} < ε, where the supremum is taken over all subsets C of R(x, B) such that for every choice k k of elements {xi}i=1 ⊂ C and scalars γi ≥ 0, i = 1,...,k with i=1 γi = 1; then k γ x ∈ R(x, B). i=1 i i P P Clearly, 1-β coincides with the property (β). Definition 5. Let k ≥ 2 be an integer. A Banach X is called to be k-nearly uniformly convex (k-NUC), provided for each ε> 0 there exists a δ,0 <δ< 1, such that for every sequence {xn} ⊂ X, with kxnk ≤ 1 and sep(xn) >ε, there are indices k k {ni}i=1 and scalars γi ≥ 0, i =1,...,k, with i=1 γi = 1 so that k P γixn ≤ 1 − δ. i=1 X Evidently, k-NUC implies NUC. 3 Theorem 6. Let X be a Banach space and k ≥ 2 be an integer. Then the following are equivalent: (i) X is k-NUC; (ii) for each ε > 0 there is a δ, 0 <δ< 1 such that for every set E contained k is the closed unit ball B with α(E) > ε, there exist elements {xi}i=1 ⊂ E so that k conv({xi}i=1) ∩ (1 − δ) =6 ∅; (iii) for each ε> 0 there is a δ, 0 <δ< 1, such that for every sequence {xn} ⊂ B k with sep(xn) >ε, there are indices {ni}i=1 so that k xni k ≤ 1 − δ. , i=1 X Proof. The equivalence between (i) and (ii) follows immediately from the properties of the Kuratowski measure of non-compactness. It remains to show that (i) ⇒ (iii). Let ε> 0 and choose δ > 0 according to Definition 5. Put η = δ/k. Let {xn} ⊂ B k be an arbitrary sequence with sep(xn) >ε. Then there exists a point y = i=1 γixni with γ ≥ 0, k γ = 1 so that kyk ≤ 1 − δ. Without loss of generality we may i i=1 i P assume that P γ1 = max{γi :1 ≤ i ≤ k}. We have the obvious representation 1 k 1 1 k γ x = y + 1 − i x . k ni k γ γ ni i=1 1 i=2 1 ! X X By the choice of γ1, 1 − γi/γ1 ≥ 0, i =2,...,k, and thus 1 k 1 δ xni ≤ k − ≤ 1 − η, k k γ1 i=1 X which concludes the proof. Theorem 7. Let X be a Banach space with closed unit ball B and k ≥ 1 be an integer. Then the following are equivalent: (i) X is k-β; (ii) for each ε> 0 there is a δ > 0 so that 1 < kxk < 1+ δ implies sup{sep(xn)} < ε, where the supremum is taken over all sequences {xn} ⊂ R(x, B) so that the convex k hull of every k elements {xni }i=1 also belongs to R(x, B); (iii) for each ε> 0 there exists a δ, 0 <δ< 1, so that for every element x ∈ B k and every sequence {xn} ⊂ B with sep(xn) > ε, there are indices {ni}i=1 so that k conv({x}∪{xni }i=1) ∩ (1 − δ)B =6 ∅; 4 (iv) for each ε> 0 there exists a δ, 0 <δ< 1, so that for every element x ∈ B n and every sequence {xn} ⊂ B with sep(xn) >ε, there are indices {ni}i=1 so that k x + xni (k + 1) ≤ 1 − δ.