A Banach space in which all compact sets, but not all bounded sets, admit Chebyshev centers

The concept of (V*) set was introduced, as a dual companion of that of (V)-set, by Pelczynski in his important paper [14]. In the same paper, the so called properties (V) and (V*) are defined by the coincidence of the (V) or (V*) sets with the weakly relatively compact sets. Many important Banach space properties are (or can be) defined in the same way; that is, by the coincidence of two classes of bounded sets. In this paper, we are concerned with the study of the class of (V*) sets in a Banach space, and its relationship with other related classes. To this general study is devoted Section I. A (as far as we know) new Banach space property (we called it property weak (V*)) is defined, by imposing the coincidence of (V*) sets and weakly conditionally compact sets. In this way, property (V*) is decomposed into the conjunction of the weak (V*) property and the weak sequential completeness. In Section II, we specialize to the study of (V*) sets in Banach lattices. The main result in the section is that every order continuous Banach lattice has property weak (V*), which extends previous results of E. and P. Saab ([16]). Finally, Section III is devoted to the study of (V*) sets in spaces of Bochner integrable functions. We characterize a broad class of (V*) sets in L1(μ, E), obtaining similar results to those of Andrews [1], Bourgain [6] and Diestel [7] for other classes of subsets. Applications to the study of properties (V*) and weak (V*) are obtained. Extension of these results to vector valued Orlicz function spaces are also given.

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Banach Spaces that are Uniformly Rotund in Weakly Compact Sets of Directions

Canadian Journal of Mathematics ◽

10.4153/cjm-1977-097-6 ◽

1977 ◽

Vol 29 (5) ◽

pp. 963-970 ◽

Cited By ~ 8

Author(s):

Mark A. Smith

Keyword(s):

Banach Space ◽

Unit Ball ◽

Banach Spaces ◽

Nonzero Element ◽

Bounded Subset ◽

Compact Sets ◽

Weakly Compact ◽

Uniform Rotundity ◽

Minimum Depth ◽

Weakly Compact Sets

In a Banach space, the directional modulus of rotundity, δ (ϵ, z), measures the minimum depth at which the midpoints of all chords of the unit ball which are parallel to z and of length at least ϵ are buried beneath the surface. A Banach space is uniformly rotund in every direction (URED) if δ (ϵ, z) is positive for every positive ϵ and every nonzero element z. This concept of directionalized uniform rotundity was introduced by Garkavi [6] to characterize those Banach spaces in which every bounded subset has at most one Čebyšev center.

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The completion of the space of convex, bounded sets with respect to the Demyanov metric

Demonstratio Mathematica ◽

10.1515/dema-2013-0433 ◽

2013 ◽

Vol 46 (1) ◽

Author(s):

Jerzy Grzybowski ◽

Andrzej Leśniewski ◽

Tadeusz Rzeżuchowski

Keyword(s):

Convex Compact ◽

Compact Sets ◽

Finite Dimensional ◽

The Family ◽

Bounded Sets ◽

Convex Compact Sets

AbstractThe Demyanov metric in the family of convex, compact sets in finite dimensional spaces has been recently extended to the family of convex, bounded sets – not necessarily closed. In this note it is shown that these spaces are not complete and a model for the completion is proposed. A full answer is given in ℝ

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Fixed Point Problems for Nonexpansive Mappings in Bounded Sets of Banach Spaces

Advances in Mathematical Physics ◽

10.1155/2020/9182016 ◽

2020 ◽

Vol 2020 ◽

pp. 1-6

Author(s):

Xianbing Wu

Keyword(s):

Banach Space ◽

Fixed Point ◽

Fixed Points ◽

Banach Spaces ◽

Iterative Process ◽

Fixed Point Theorems ◽

Nonexpansive Mappings ◽

Fixed Point Problems ◽

Bounded Sets

It is well known that nonexpansive mappings do not always have fixed points for bounded sets in Banach space. The purpose of this paper is to establish fixed point theorems of nonexpansive mappings for bounded sets in Banach spaces. We study the existence of fixed points for nonexpansive mappings in bounded sets, and we present the iterative process to approximate fixed points. Some examples are given to support our results.

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Mazur's intersection property of balls for compact convex sets

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700013228 ◽

1987 ◽

Vol 35 (2) ◽

pp. 267-274 ◽

Cited By ~ 4

Author(s):

J. H. M. Whitfield ◽

V. Zizler

Keyword(s):

Banach Space ◽

Uniform Convergence ◽

Convex Sets ◽

Convex Set ◽

Compact Convex ◽

Extreme Points ◽

Intersection Property ◽

Compact Sets ◽

Compact Convex Set ◽

Mazur's Intersection Property

We show that every compact convex set in a Banach space X is an intersection of balls provided the cone generated by the set of all extreme points of the dual unit ball of X* is dense in X* in the topology of uniform convergence on compact sets in X. This allows us to renorm every Banach space with transfinite Schauder basis by a norm which shares the mentioned intersection property.

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Farthest points and the farthest distance map

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700038430 ◽

2005 ◽

Vol 71 (3) ◽

pp. 425-433 ◽

Cited By ~ 4

Author(s):

Pradipta Bandyopadhyay ◽

S. Dutta

Keyword(s):

Banach Space ◽

Maximal Monotone ◽

Convex Banach Space ◽

Closed Convex Hull ◽

Compact Sets ◽

Weakly Compact ◽

Distance Map ◽

Farthest Points ◽

Bounded Set ◽

Weakly Compact Sets

In this paper, we consider farthest points and the farthest distance map of a closed bounded set in a Banach space. We show, inter alia, that a strictly convex Banach space has the Mazur intersection property for weakly compact sets if and only if every such set is the closed convex hull of its farthest points, and recapture a classical result of Lau in a broader set-up. We obtain an expression for the subdifferential of the farthest distance map in the spirit of Preiss' Theorem which in turn extends a result of Westphal and Schwartz, showing that the subdifferential of the farthest distance map is the unique maximal monotone extension of a densely defined monotone operator involving the duality map and the farthest point map.

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Lines and Hyperplanes associated with Families of Closed and Bounded Sets in Conjugate Banach Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-1970-107-3 ◽

1970 ◽

Vol 22 (5) ◽

pp. 933-938

Author(s):

M. Edelstein

Keyword(s):

General Setting ◽

Locally Convex Space ◽

Finite Family ◽

Topological Spaces ◽

Compact Sets ◽

Straight Line ◽

Straight Lines ◽

Bounded Sets ◽

Infinite Set ◽

Locally Convex

Let be a family of sets in a linear space X. A hyperplane π is called a k-secant of if π intersects exactly k members of . The existence of k-secants for families of compact sets in linear topological spaces has been discussed in a number of recent papers (cf. [3–7]). For X normed (and a finite family of two or more disjoint non-empty compact sets) it was proved [5] that if the union of all members of is an infinite set which is not contained in any straight line of X, then has a 2-secant. This result and related ones concerning intersections of members of by straight lines have since been extended in [4] to the more general setting of a Hausdorff locally convex space.

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