Orlicz Spaces Without Strongly Extreme Points and Without <i>H</i>-Points

Hudzik, Henryk

doi:10.4153/cmb-1993-025-6

Cited by 7 publications

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“…Stronger points than extreme points of B( X) are exposed points, rotund points, strongly extreme points, strongly exposed points, denting points and LUR-points (see [5,9,12,14,16,19,22,26]). …”

Section: Introduction and Auxiliary Resultsmentioning

confidence: 99%

Orlicz spaces without extreme points

Foralewski¹,

Hudzik²,

Płuciennik³

2010

Journal of Mathematical Analysis and Applications

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Orlicz function and sequence spaces unit balls of which have no extreme points are completely characterized for both (the Orlicz and the Luxemburg) norms. Their subspaces of order continuous elements, with the norms induced from the whole Orlicz spaces without extreme points in their unit balls are also characterized. The well-known spaces L 1 and c 0 with unit balls without extreme points are covered by our results. Moreover, a new example of a Banach space without extreme points in its unit ball is given (see Example 1). This is the subspace (L 1 + L ∞ ) a of order continuous elements of the space L 1 + L ∞ equipped with the norm x = a 1/a 0 x * (t) dt whenever 0 < a < ∞ and μ(T ) > 1/a.

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Section: Introduction and Auxiliary Resultsmentioning

confidence: 99%

Orlicz spaces without extreme points

Foralewski¹,

Hudzik²,

Płuciennik³

2010

Journal of Mathematical Analysis and Applications

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“…If the measure of the underlying space is nonatomic, then A(Lp) --(1 + 21-1/P) -~ for 1 < p __ 2 and A(Lp) = (l+2~/P) -x for 2 <_ p < co [4]. The relationship between the packing constant of a space and its reflexivity was studied in [1,5]. Upper and lower estimates for the packing constant in Orlicz spaces were obtained in [6][7][8][9].…”

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confidence: 99%

The packing constant in rearrangement-invariant spaces

Appell¹,

Semenov²

1998

Funct Anal Its Appl

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Let E be an infinite-dimensional Banach space and B(x, r) the closed ball of radius r and center x in E. The number { )A(E) = sup r: r > 0, 3xi e E, i = 1,2,...,"Ilxi -zsll > 2~, B(z~, ~) r B(0, l)is called the packing constant of E. Likewise, the finite packing constant A s (E) is defined byObviously, A(E) _< Af(E). It was shown in [1] thatwhereThe finite analog of D(E) isThe constant A(/2) was calculated in [2]. This result was extended to lp in [3]: A(/p) --(1 + 21-1/P) -1 for 1 < p < oc. If the measure of the underlying space is nonatomic, then A(Lp) --(1 + 21-1/P) -~ for 1 < p __ 2 and A(Lp) = (l+2~/P) -x for 2 <_ p < co [4]. The relationship between the packing constant of a space and its reflexivity was studied in [1,5]. Upper and lower estimates for the packing constant in Orlicz spaces were obtained in [6][7][8][9]. The minimum radius of a ball that contains some elements xl, ..., x~ E lp with [Ixi -xjN >_ 1 for 1 _< i • j < n was found in [10]. For any infinite-dimensional space E, one has D(E) > 1 [11]; consequently, 1/3 < A(E) _< 1/2.In this paper, we study the packing constant in rearrangement-invariant spaces. A Banach space E of measurable functions on [0, 1] is said to be rearrangement-invariant, or symmetric, if x* (t) < y* (t) with y E E implies that x E E and [[xiig _< [[YiiE, where x*(t) sup x*(t)t 1/p, q = oo,

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On Some Convexity Properties of Orlicz Sequence Spaces Equipped with the Luxemburg Norm

Hudzik

Pallaschke

1997

Mathematische Nachrichten

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Rotundity of finite-diii~eilsioiial Orlin spaces 1: equipped with the Luxemburg nomi is considered. It is proved that criteria for rotundity of 1: for 11 2 3 does not depend on 11 and are the same as the criteria for rotundity of the inhite-dimensional subspace h* of an Orlicz sequence ~p a c e . 1~. Criteria for rotundity of 1; are different. Next, criteria for exposed points, (H)-points, strongly exposed points and LUR-points of the unit sphere of 1 ' and of its subspace h* are given. I" =; { z = ( x i ) E 1' : Z*(AZ) 4 +m for some A > 0) i = l 1991 Maihtmaiics Subjeci Clarsificaiion. Primary 46330; Secondary 46B 20. Keywords and phrarer. Orlicz space, Luxemburg norm, extreme point, exposed point, strongly -xpored point, (H)-point, LUR-point, (H)-property of compact convex acts. Lemma 2.1. Let 0 be an arbitrary Orlicz function. Then IIzllq,= 1 is equivalent to /m(c)= 1 for any t E ha if and only i f @ ( b ( @ ) ) z 1.

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Orlicz Spaces Without Strongly Extreme Points and Without H-Points

Cited by 7 publications

References 5 publications

Orlicz spaces without extreme points

Orlicz spaces without extreme points

The packing constant in rearrangement-invariant spaces

On Some Convexity Properties of Orlicz Sequence Spaces Equipped with the Luxemburg Norm

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