一致凸度量线性空间的自反性及其应用

为研究度量线性空间中凸集的逼近性质，Ｇ．Ｃ．Ａｈｕｊａ等引起了度量线性空间的严格凸性及一致凸性的定义。本文证明了完备的一致凸的度量线性空间是自反的。同时，作为应用，研究了最佳联合逼近元的存在性与唯一性问题。     

凸性与度量投影的连续性

本文研究近强凸、近非常凸Banach空间中度量投影的连续性。获得如下结果：若A是近强凸（近非常凸）空间中的逼近凸集,则度量投影PA是范-范上半连续的（范-弱上半连续的）。此外,我们还利用度量投影的连续性给出Banach空间为近强凸、近非常凸的一些充分必要条件。     

Giuseppe Tallini: His life and work

It is known that euclidean or hyperbolic spaces are characterized among certain metric spaces by the property of linearity of the equidistant locus of pairs of points. In this paper, this linearity requirement is replaced by the requirement of convexity of the set of points which are metrically pythagorean orthogonal to a given segment at a given point. As a result a new characterization of real inner product spaces among complete, convex, externally convex metric spaces is obtained.     

关于度量投影的连续性

本文引入的Ｂａｎａｃｈ空间的（Ｃ－Ｉ）、（Ｃ－Ⅱ），（Ｃ－Ⅲ）等几何性质，证明了如下结果。设Ｍ是Ｂａｎａｃｈ空间的逼近凸子集，如果Ｂａｎａｃｈ空间有性质（Ｃ－Ｉ），（Ｃ－Ⅱ）（Ｃ－Ⅲ），则度量投影ＰＭ连续（范数－范数上半连续，范数－弱上半连续）。这些结果推广了文（４，７，８）相应的定理。最近，Ｄ．Ｋｕｔｚａｒｏｖａ，Ｂｏｒ－Ｌｕｈ Ｌｉｎ等引入了一些新的凸性空间，本文还研究了这些凸性空间中度量投影     

Characterization of inner product spaces in V- and L-spaces

In [4], Freese and Murphy introduce a new class of spaces, the V-spaces, which include Banach spaces, hyperbolic spaces, and other metric spaces. In this class of spaces they investigate conditions which are equivalent to strict convexity in Banach spaces, and extend some of the Banach space results to this new class of spaces. It is natural to ask if known characterizations of real inner product spaces among Banach spaces can also be extended to this larger class of spaces. In the present paper it will be shown that a metrization of an angle bisector property used in [3] to characterize real inner product spaces among Banach spaces also characterizes real inner product spaces among V-spaces, and among another class of spaces, the L-spaces, which include hyperbolic spaces and strictly convex Banach spaces. In the process it is shown that in a complete, convex, externally convex metric space M, if the foot of a point on a metric line is unique, then M satisfies the monotone property, thus answering a question raised in [4].     

Topological convexities, selections and fixed points

A convexity on a set X is a family of subsets of X which contains the whole space and the empty set as well as the singletons and which is closed under arbitrary intersections and updirected unions. A uniform convex space is a uniform topological space endowed with a convexity for which the convex hull operator is uniformly continuous. Uniform convex spaces with homotopically trivial polytopes (convex hulls of finite sets) are absolute extensors for the class of metric spaces; if they are completely metrizable then a continuous selection theorem à la Michael holds. Upper semicontinuous maps have approximate selections and fixed points, under the usual assumptions.     

Existence and convergence theorems for generalized hybrid mappings in uniformly convex metric spaces

In this paper, we show a relationship between strictly convexity of type (I) and (II) defined by Takahashi and Talman, and we prove that any uniformly convex metric space is strictly convex of type (II). Continuity of the convex structure is also shown on a compact domain. Then, we prove the existence of a minimum point of a convex, lower semicontinuous and d-coercive function defined on a nonempty closed convex subset of a complete uniformly convex metric space. By using this property, we prove fixed point theorems for (α, β)-generalized hybrid mappings in uniformly convex metric spaces. Using this result, we also obtain a common fixed point theorem for a countable commutative family of (α, β)-generalized hybrid mappings in uniformly convex metric spaces. Finally, we establish strong convergence of a Mann type iteration to a fixed point of (α, β)-generalized hybrid mapping in a uniformly convex metric space without assuming continuity of convex structure. Our results can be applied to obtain the existence and convergence theorems for (α, β)-generalized hybrid mappings in Hilbert spaces, uniformly convex Banach spaces and CAT(0) spaces.     

The Knaster-Kuratowski-Mazurkiewicz theorem and abstract convexities

The Knaster-Kuratowski-Mazurkiewicz covering theorem (KKM), is the basic ingredient in the proofs of many so-called “intersection” theorems and related fixed point theorems (including the famous Brouwer fixed point theorem). The KKM theorem was extended from Rn to Hausdorff linear spaces by Ky Fan. There has subsequently been a plethora of attempts at extending the KKM type results to arbitrary topological spaces. Virtually all these involve the introduction of some sort of abstract convexity structure for a topological space, among others we could mention H-spaces and G-spaces. We have introduced a new abstract convexity structure that generalizes the concept of a metric space with a convex structure, introduced by E. Michael in [E. Michael, Convex structures and continuous selections, Canad. J. Math. 11 (1959) 556-575] and called a topological space endowed with this structure an M-space. In an article by Shie Park and Hoonjoo Kim [S. Park, H. Kim, Coincidence theorems for admissible multifunctions on generalized convex spaces, J. Math. Anal. Appl. 197 (1996) 173-187], the concepts of G-spaces and metric spaces with Michael's convex structure, were mentioned together but no kind of relationship was shown. In this article, we prove that G-spaces and M-spaces are close related. We also introduce here the concept of an L-space, which is inspired in the MC-spaces of J.V. Llinares [J.V. Llinares, Unified treatment of the problem of existence of maximal elements in binary relations: A characterization, J. Math. Econom. 29 (1998) 285-302], and establish relationships between the convexities of these spaces with the spaces previously mentioned.     

Uniformly convex spaces,bead spaces,and equivalence conditions

The notion of a metric bead space was introduced in the preceding paper (L.Pasicki: Bead spaces and fixed point theorems, Topology Appl., vol. 156 (2009), 1811–1816) and it was proved there that every bounded set in such a space (provided the space is complete) has a unique central point. The bead spaces themselves can be considered in particular as natural extensions of convex sets in uniformly convex spaces. It appears that normed bead spaces are identical with uniformly convex spaces. On the other hand the “metric” approach leads to new elementary conditions equivalent to the uniform convexity. The initial part of the paper contains the proof that discus spaces (they seem to have a richer structure) are identical with bead spaces.     

Strong additivity of metric isosceles orthogonality

It is known that the property of additivity of isosceles orthogonality characterizes real inner product spaces among normed linear spaces. In the present paper it is shown that suitably metrized concepts of additivity of metric isosceles orthogonality characterize euclidean or hyperbolic spaces among complete, convex, externally convex metric spaces.     

线性距离空间的一致凸性与自反性

本文研究了线性距离空间的一致凸性与自反性，同时对吴从Ｘｉｎ等提出的严格缩条件能否去掉的问题给出了肯定的答案。     

Weak additivity of metric pythagorean orthogonality

It is well known that the property of additivity of pythagorean orthogonality characterizes real inner product spaces among normed linear spaces. In the present paper, a natural concept of additivity is introduced in metric spaces, and it is shown that a weakened version of this additivity of metric pythagorean orthogonality characterizes real inner product spaces among complete, convex, externally convex metric spaces, providing a generalization of the earlier characterization.     

局部凸空间的k-一致极凸性与k-一致极光滑性

首先引入局部凸空间的k-一致极凸性和k-一致极光滑性这一对对偶概念,它们既是Banach空间k-一致极凸性和k-一致极光滑性推广,又是局部凸空间一致极凸性和一致极光滑性的自然推广.其次讨论它们与其它k-凸性(k-光滑性)之间的关系.最后,在P-自反的条件下给出它们之间的等价对偶定理.     

Pythagorean euclidean four point properties

Characterizations of real inner product spaces among a class of metric spaces have been obtained based on homogeneity of metric pythagorean orthogonality, a metrization of the concept of pythagorean orthogonality as defined in normed linear spaces. In the present paper a considerable weakening of this hypothesis is shown to characterize real inner product spaces among complete, convex, externally convex metric spaces, generalizing a result of Kapoor and Prasad [9], and providing a connection with the many characterizations of such spaces using euclidean four point properties.     

Convexities of metric spaces

We introduce two kinds of the notion of convexity of a metric space, called k-convexity and L-convexity, as generalizations of the CAT(0)-property and of the nonpositively curved property in the sense of Busemann, respectively. 2-uniformly convex Banach spaces as well as CAT(1)-spaces with small diameters satisfy both these convexities. Among several geometric and analytic results, we prove the solvability of the Dirichlet problem for maps into a wide class of metric spaces.      

Warped products of Hadamard spaces

In the Riemannian case, our approach to warped products illuminates curvature formulas that previously seemed formal and somewhat mysterious. Moreover, the geometric approach allows us to study warped products in a much more general class of spaces. For complete metric spaces, it is known that nonpositive curvature in the Alexandrov sense is preserved by gluing on isometric closed convex subsets and by Gromov–Hausdorff limits with strictly positive convexity radius; we show it is also preserved by warped products with convex warping functions. Received: 9 January 1998/ Revised version: 12 March 1998     

Nearly strict convexity in Musielak-Orlicz-Bochner function spaces

Criteria for nearly strict convexity of Musielak-Orlicz-Bochner function spaces equipped with the Luxemburg norm are given. We also prove that, in Musielak-Orlicz-Bochner function spaces generated by strictly convex Banach space, nearly strict convexity and strict convexity are equivalent.     

A decomposition theorem for convexity spaces

We utilize the unifying framework of families of convexity spaces for the treatment of various notions of planar convexity and the associated convex hulls. Our major goal is to prove the refinement and decomposition theorems for families of convexity spaces. These general theorems are then applied to two examples: restricted-oriented convex sets andNESW-convex sets. The applications demonstrate the usefulness of these general theorems, since they give rise to simple algorithms for the computation of the associated convex hulls of polygons.     

Locally uniform convexity in Musielak-Orlicz function spaces of Bochner type endowed with the Luxemburg norm

Criteria for locally uniform convexity of Musielak-Orlicz function spaces of Bochner type equipped with the Luxemburg norm are given. We also prove that, in Musielak-Orlicz function spaces generated by locally uniformly convex Banach space, locally uniform convexity and strict convexity are equivalent.     

The Metric Completion of Convex Sets and Modules

A convex module is a set closed under convex combinations but not necessarily a subset of a linear space. Convex modules are a natural generalization of convex sets in linear spaces. Any-convex module has a canonical semimetric and there is a universal affine mapping into a convex set, complete in this metric. This completion is the base of a base normed Banach space.