Extreme points and retractions in Banach spaces

ForT a completely regular topological space andX a strictly convex Banach space, we study the extremal structure of the unit ball of the spaceC(T,X) of continuous and bounded functions fromT intoX. We show that when dimX is an even integer then every point in the unit ball ofC(T, X) can be expressed as the average of three extreme points if, and only if, dimT< dimX, where dimT is the covering dimension ofT. We also prove that, ifX is infinite-dimensional, the aforementioned representation of the points in the unit ball ofC(T, X) is always possible without restrictions on the topological spaceT. Finally, we deduce from the above result that the identity mapping on the unit ball of an infinite-dimensional strictly convex Banach space admits a representation as the mean of three retractions of the unit ball onto the unit sphere. The author wishes to express his gratitude to Dr. Juan Francisco Mena Jurado for many helpful suggestions during the preparation of this paper.     

Best approximation in polyhedral Banach spaces

In the present paper, we study conditions under which the metric projection of a polyhedral Banach space X onto a closed subspace is Hausdorff lower or upper semicontinuous. For example, we prove that if X satisfies (∗) (a geometric property stronger than polyhedrality) and Y⊂X is any proximinal subspace, then the metric projection P_Y is Hausdorff continuous and Y is strongly proximinal (i.e., if {y_n}⊂Y, x∈X and , then ).One of the main results of a different nature is the following: if X satisfies (∗) and Y⊂X is a closed subspace of finite codimension, then the following conditions are equivalent: (a) Y is strongly proximinal; (b) Y is proximinal; (c) each element of Y^⊥ attains its norm. Moreover, in this case the quotient X/Y is polyhedral.The final part of the paper contains examples illustrating the importance of some hypotheses in our main results.     

Three characterizations of polyhedral Banach spaces

Three new criteria for the polyhedrality of a Banach space are given in terms of norms, locally dependent on a finite number of coordinates, tilings of the space, and coverings of the sphere by a sequence of cuts.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 9, pp. 1286–1290, September, 1990.     

Analytic and polyhedral approximation of convex bodies in separable polyhedral Banach spaces

A closed, convex and bounded setP in a Banach spaceE is called a polytope if every finite-dimensional section ofP is a polytope. A Banach spaceE is called polyhedral ifE has an equivalent norm such that its unit ball is a polytope. We prove here:

Strongly extreme points in Banach spaces

In this paper some properties of a special type of boundary point of convex sets in Banach spaces are studied. Specifically, a strongly extreme point x of a convex set S is a point of S such that for each real number r>0, segments of length 2r and centered x are not uniformly closer to S than some positive number d(x,r). Results are obtained comparing the notion of strongly extreme point to other known types of special boundary points of convex sets. Using the notion of strongly extreme point, a convexity condition is defined on the norm of the space under consideration, and this convexity condition makes possible a unified treatment of some previously studied convexity conditions. In addition, a sufficient condition is given on the norm of a separable conjugate space for every extreme point of the unit ball to be strongly extreme.     

Dentability and extreme points in Banach spaces

It is shown that a Banach space E has the Radon-Nikodym property (equivalently, every bounded subset of E is dentable) if and only if every bounded closed convex subset of E is the closed convex hull of its strongly exposed points. Using recent work of Namioka, some analogous results are obtained concerning weak¹ strongly exposed points of weak¹ compact convex subsets of certain dual Banach spaces.     

Locally uniformly rotund points in convex-transitive Banach spaces

Almost transitive superreflexive Banach spaces have been considered in [C. Finet, Uniform convexity properties of norms on superreflexive Banach spaces, Israel J. Math. 53 (1986) 81–92], where it is shown that they are uniformly convex and uniformly smooth. We characterize such spaces as those convex transitive Banach spaces satisfying conditions much weaker than that of uniform convexity (for example, that of having a weakly locally uniformly rotund point). We note that, in general, the property of convex transitivity for a Banach space is weaker than that of almost transitivity.     

Extreme points of the set of Banach limits

Let ${\mathbb{N}}$ be the set of all natural numbers and ${\ell_\infty=\ell_\infty (\mathbb{N})}$ be the Banach space of all bounded sequences x = (x ₁, x ₂ . . .) with the norm $$\|x\|_{\infty}=\sup_{n\in\mathbb{N}}|x_n|,$$ and let ${\ell_\infty^*}$ be its Banach dual. Let ${\mathfrak{B} \subset \ell_\infty^*}$ be the set of all normalised positive translation invariant functionals (Banach limits) on ? _∞ and let ${ext(\mathfrak{B})}$ be the set of all extreme points of ${\mathfrak{B}}$ . We prove that an arbitrary sequence (B _j ) _{j ≥ 1}, of distinct points from the set ${ext(\mathfrak{B})}$ is 1-equivalent to the unit vector basis of the space ? ₁ of all summable sequences. We also study Cesáro-invariant Banach limits. In particular, we prove that the norm closed convex hull of ${ext(\mathfrak{B})}$ does not contain a Cesáro-invariant Banach limit.     

Fixed points of asymptotically regular semigroups in Banach spaces

In this paper we study in Banach spaces the existence of fixed points of (nonlinear) asymptotically regular semigroups. We establish for these semigroups some fixed point theorems in spaces with weak uniform normal structure, in a Hilbert space, inL ^p spaces, in Hardy spacesH ^p and in Sobolev spacesW ^r.p for 1<p<∞ andr≥0, in spaces with Lifshitz’s constant greater than one. These results are the generalizations of [8, 10, 16].     

Fixed points for weakly inward mappings in Banach spaces

S. Hu and Y. Sun [S. Hu, Y. Sun, Fixed point index for weakly inward mappings, J. Math. Anal. Appl. 172 (1993) 266-273] defined the fixed point index for weakly inward mappings, investigated its properties and studied the fixed points for such mappings. In this paper, following S. Hu and Y. Sun, we continue to investigate boundary conditions, under which the fixed point index for the completely continuous and weakly inward mapping, denoted by i(A,Ω,P), is equal to 1 or 0. Correspondingly, we can obtain some new fixed point theorems of the completely continuous and weakly inward mappings and existence theorems of solutions for the equations Ax=μx, which extend many famous theorems such as Leray-Schauder's theorem, Rothe's two theorems, Krasnoselskii's theorem, Altman's theorem, Petryshyn's theorem, etc., to the case of weakly inward mappings. In addition, our conclusions and methods are different from the ones in many recent works.     

Extreme point properties of convex bodies in reflexive Banach spaces

A study is made of topological and cardinality properties of the set of extreme points of bounded closed convex sets with interior in reflexive Banach spaces. Some related results, and applications to earlier work, are obtained as corollaries. The work on this paper was supported by the National Science Foundation, and was performed while the first named author was visiting at the University of Washington.     

Extreme points of Banach lattices related to conditional expectations

Let (X,F,μ) be a complete probability space, B a sub-σ-algebra, and Φ the probabilistic conditional expectation operator determined by B. Let K be the Banach lattice {f∈L¹(X,F,μ):‖Φ(|f|)_‖∞<∞} with the norm ‖f‖=‖Φ(|f|)_‖∞. We prove the following theorems:

(1)

The closed unit ball of K contains an extreme point if and only if there is a localizing set E for B such that supp(Φ(χE))=X.

(2)

Suppose that there is n∈N such that f?nΦ(f) for all positive f in L^∞(X,F,μ). Then K has the uniformly λ-property and every element f in the complex K with is a convex combination of at most 2n extreme points in the closed unit ball of K.

     

Farthest points in reflexive locally uniformly rotund Banach spaces

IfS is a bounded and closed subset of a Banach spaceB, which is both reflexive and locally uniformly rotund, then, except on a set of first Baire category, the points inB have farthest points inS.     

Fixed points of asymptotically regular semigroups in Banach spaces

In this paper we study in Banach spaces the existence of fixed points of (nonlinear) asymptotically regular semigroups. We establish for these semigroups some fixed point theorems in spaces with weak uniform normal structure, in a Hilbert space, inL ^p spaces, in Hardy spacesH ^p and in Sobolev spacesW ^r.p for 1<p<∞ andr≥0, in spaces with Lifshitz’s constant greater than one. These results are the generalizations of [8, 10, 16].