Bor-Luh Lin

On the K-uniform rotund and the fully convex Banach spaces

Let X be a real Banach space. If X is uniformly convex then it is known that X is 2-uniformly rotund [S] and fully 2-convex [2]. Furthermore, if a Banach space X is either k-uniformly rotund or fully k-convex for some k, then X is reflexive. In this paper, we show that if X is strictly convex and kuniformly rotund, then X is fully (k + 1)-convex. However, there exists a superreflexive space which is fully 2-convex but is not 2-uniformly rotund and for each k > 2, there exists a strictly convex space which is k-uniformly rotund but is not fully k-convex. Thus for each k > 2, there exist fully kconvex Banach space which is not fully (k 1)-convex. For x,, x2 ,..., xk+ , in X, let

Property

We prove that if a Banach space X has the property (HR) and if /, is not isomorphic to a subspace of X, then every point on the unit sphere of X is a denting point of the closed unit ball. We also prove that if X has the above property, then Lp(p, X), 1 < p < oo, has the property (H).

(H)

Let k 2 2 be an integer. A Banach space X is said to be fully k-convex (kR) if for every sequence {xn> in X, lim,,,,..,n,, o. (l/k) II&l, + ... + x,,J = 1 implies that {xn} is a Cauchy sequence in X. It is known [S] that every uniformly convex space is 2R and that every kR space is reflexive. It is also known [3, 51 that there are 2R spaces which are not superreflexive. Milman [ 10, p. 973 has raised the question whether every reflexive space is isomorphic to a 2R space. A Banach space X is said to have the Banach-Saks property (BS) if for every bounded sequence {x,} in X, there exists a subsequence {y,} of (xn} such that the Cesaro means, { (l/n)( y, + . . . + y,}, of { yn} is convergent in X. It is known that [7] all superreflexive spaces have the (BS) and that [ 1 l] every Banach space with (BS) is reflexive. However, there exist (e.g., [1]) reflexive Banach spaces which do not have the (BS). It is also known [12] that there are Banach spaces with the (BS) which are not superreflexive. Since the (BS) is invariant under isomorphism, it is natural to ask whether every kR space has the (BS). An affirmative answer would yield a negative solution to Milman’s question. The purpose of the paper is to renorm Baernstein’s space [ 1 ] so that it is 2R. Therefore, kR spaces do not necessarily have the (BS). We shall need the following characterization of kR spaces. For the proof when k = 2, see [S, 6, S].

in Lebesgue-Bochner function spaces

We show that in Orlicz function spaces with Orlicz/Luxemburg norm the criteria for being noncreasy and uniformly noncreasy are interesting combinations of conditions.

A fully convex Banach space which does not have the Banach-Saks property

DEFINITION. Let Y be a family of closed convex subsets of a Banach space X. For a real number, (Y, we shall say X possesses property D(or, 9) if for each ,4 E 9 and each ICE X there is a pomt u E X such that x E u + aPA( (We shall use the convention that u + o = o .) Nashed, in the paper cited above, showed that every Hilbert space possesses D( 1, YO) where Y0 is the famdy of all closed nonempty convex subsets of the space. Our investigation is divided into three sections which consider, respectively, the cases when (Y > 1, 01 = 1, and ar < 1. In each case we obtain a characterization of reflexivity (Theorem 1, Theorem 2, and