On a convexity condition in normed linear spaces

Abstract

Introduction. The purpose of this paper is to study a convexity property on normed linear spaces (NLS's) which we call P-convexity. Interest in P-convexity is generated by a theorem of Anatole Beck ([1] or [2]) which states that a Banach space 3£ is P-convex if and only if a certain strong law of large numbers is valid for X-valued random variables. Let 3£ be a NLS and (S, 2, m) a measure space. The Borel a-field 38 of X is the tr-field of sets generated by the subsets of BE open in the strong (norm) topology. A Borel set is an element of 38. A function X from 5 into 3t is called strongly measurable if for each Be Sä, X~ \B) = {se S : X(s) e B} e 2. X is called essentially separably valued if there is N eH, m(N) = 0, such that X(S-N) is separable. If js || X(s) || dm{s) < oo and X is essentially separably valued, then X is strongly (Bochner) integrable and there exists ye?), where ?) is the completion of 3£, such that for every x* e £)*, the conjugate space to 9), Js x*X(s) dm(s) = x*y. In this case we write y=js X(s) dm(s)=$s X dm. y is called the strong integral of X. A probability space is a measure space (Í2, 2, 3P) of total measure 1 (^(D)= 1). An 3t-valued random variable on Q. is a function X: Q.-*■ X which is strongly measurable and essentially separably valued. The expectation of X, E{X) = ja X d£? if this integral exists. For a random variable X with expectation, we define the variance of X by (20= f ixw-EvnvtWr). Ja var i We will call a finite set Xu..., Xn of 3t-valued random variables on Q. independent if for each choice of Borel sets Blt..., Bn e 38, we have d\Xi eB1&-&XneBn) = 3^{XX eB,)-3P{Xn e Bn); that is, n {o £ Q : XiUm) e B1 & • ■ & Xn(oj) e Bn) = Y\0>{ojeQ.: X¿w) e Bt}. i = i An infinite set of random variables is independent if each finite subset is.

𝑃-convexity and 𝐵-convexity in Banach spaces

“…In this paper we consider two of these properties and prove partial analogs of them for P-convex spaces: The first property is that direct sums of B-convex spaces are B-convex [2]. The proof of this fact for B-convex spaces rests on the invariance of B-convexity under isomorphism, but it is not known whether P-convexity possesses this invariance.…”

“…Then either (1) there is A C S, a subset with p elements, so that any r subset of A is in a, or (2) there is B C S, a subset of q elements, so that any r subset of B is in ß.…”

Rotundity in Lebesgue-Bochner function spaces

“…Using a result of Giesy [9] which states that X is not 5-convex if and only if / ' is finitely representable in X, Figiel [17] proved the following corollary. The proof of this result is immediate from Theorem 15.…”

P-Convexity Property in Musielak-Orlicz Function Space of Bohner Type

“…Yining, et.al [5,6] in their paper entitled "P-convexity and reflexivity of Orlicz spaces" proved that for Orlicz spaces reflexivity is equivalent to Pconvexity. The same result for the MusielakOrlicz sequence and function spaces were obtained by Kolwicz and Pluciennik [7][8][9].…”