Orlicz空间的若干紧一致凸性和k-drop凸性

给出赋Luxemburg范数的Orlicz函数空间的紧一致凸、弱紧一致凸、紧局部一致凸、弱紧局部一致凸和k-drop凸的判据,并且据此得到在Orlicz函数空间中这些凸性的等价关系.     

Geodesic convexity in nonlinear optimization

The properties of geodesic convex functions defined on a connected RiemannianC ² k-manifold are investigated in order to extend some results of convex optimization problems to nonlinear ones, whose feasible region is given by equalities and by inequalities and is a subset of a nonlinear space.This research was supported in part by the Hungarian National Research Foundation, Grant No. OTKA-1044.     

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.     

Local convexity properties of quasihyperbolic balls in punctured space

This paper deals with local convexity properties of the quasihyperbolic metric in the punctured space. We consider convexity and starlikeness of the quasihyperbolic balls.     

Takagi functions and approximate midconvexity

Let V be a convex subset of a normed space and let ε?0, p>0 be given constants. A function f:V→R is called (ε,p)-midconvex if

     

Improving constants of strong convexity in linear stochastic programming

We derive formulas for constants of strong convexity (CSCs) of expectation functions encountered in two-stage stochastic programs with linear recourse. One of them yields a CSC as the optimal value of a certain quadratically constrained quadratic program, another one in terms of the thickness of the feasibility polytope of the dual problem associated to the recourse problem. CSCs appear in Hoelder-type estimates relating the distance of optimal solution sets of stochastic programs to a suitable distance of underlying probability distributions.     

关于凸性特征与光滑模的一个关系

给出了Banach空间凸性特征与光滑模的一个关系。     

The Krein-Milman theorem in operator convexity

We generalize the Krein-Milman theorem to the setting of matrix convex sets of Effros-Winkler, extending the work of Farenick-Morenz on compact C-convex sets of complex matrices and the matrix state spaces of C-algebras. An essential ingredient is to prove the non-commutative analogue of the fact that a compact convex set may be thought of as the state space of the space of continuous affine functions on .

     

The Bishop-Phelps-Bollobás theorem for operators

We prove the Bishop-Phelps-Bollobás theorem for operators from an arbitrary Banach space X into a Banach space Y whenever the range space has property β of Lindenstrauss. We also characterize those Banach spaces Y for which the Bishop-Phelps-Bollobás theorem holds for operators from ?₁ into Y. Several examples of classes of such spaces are provided. For instance, the Bishop-Phelps-Bollobás theorem holds when the range space is finite-dimensional, an L₁(μ)-space for a σ-finite measure μ, a C(K)-space for a compact Hausdorff space K, or a uniformly convex Banach space.     

关于ω-强凸空间的一点注记

本文研究了关于ω-强凸空间和ω-强光滑空间的问题.利用Banach理论的方法,证明了ω-强凸空间和ω-强光滑空间是一对对偶概念,并讨论了ω-强光滑性与其它光滑性之间的关系,用切片统一刻画了ω-强凸空间与ω-强光滑空间的特征,完善了ω-强凸空间及其对偶空间的研究.     

Clarkson不等式与Banach空间几何

我们证明了Banach空间X是Clarkson p型（q余型）当且仅当X是一个特殊的p一致光滑空间（q-一致凸空间（）,我们还找到刻划型（余型）的一系列鞅不等式,同时,我们得到了均方函数sharp不等式。     

Weakly convex sets and modulus of nonconvexity

We consider a definition of a weakly convex set which is a generalization of the notion of a weakly convex set in the sense of Vial and a proximally smooth set in the sense of Clarke, from the case of the Hilbert space to a class of Banach spaces with the modulus of convexity of the second order. Using the new definition of the weakly convex set with the given modulus of nonconvexity we prove a new retraction theorem and we obtain new results about continuity of the intersection of two continuous set-valued mappings (one of which has nonconvex images) and new affirmative solutions of the splitting problem for selections. We also investigate relationship between the new definition and the definition of a proximally smooth set and a smooth set.     

Characterizations of generalized convexities via generalized directional derivatives

The paper gives characterizations of convexity, quasiconvexity, invexity and pseudoconvexity for a (radially) upper-semicontinuous function f in a topological vector space via appropriate properties of a bifunction which is majorized by the upper radial derivative of f and which stands for a generalized derivative of some sort.     

A further characteristic of abstract convexity structures on topological spaces

In this paper, we give a characteristic of abstract convexity structures on topological spaces with selection property. We show that if a convexity structure C defined on a topological space has the weak selection property then C satisfies H₀-condition. Moreover, in a compact convex subset of a topological space with convexity structure, the weak selection property implies the fixed point property.     

On uniformly convex functions

Non-convex functions that yet satisfy a condition of uniform convexity for non-close points can arise in discrete constructions. We prove that this sort of discrete uniform convexity is inherited by the convex envelope, which is the key to obtain other remarkable properties such as the coercivity. Our techniques allow to retrieve Enflo's uniformly convex renorming of super-reflexive Banach spaces as the regularization of a raw function built from trees. Among other applications, we provide a sharp estimation of the distance of a given function to the set of differences of Lipschitz convex functions. Finally, we prove the equivalence of several possible ways to quantify the super weakly noncompactness of a convex subset of a Banach space.     

Nonpositive curvature in p-Schatten class

We study the geometry of the set Δp, with 1<p<∞, which consists of perturbations of the identity operator by p-Schatten class operators, which are positive and invertible as elements of B(H). These manifolds have natural and invariant Finsler structures. In [C. Conde, Geometric interpolation in p-Schatten class, J. Math. Anal. Appl. 340 (2008) 920-931], we introduced the metric dp and exposed several results about this metric space. The aim of this work is to prove that the space (Δp,dp) behaves in many senses like a nonpositive curvature metric space.     

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.     

Orthogonally Complemented Subspaces in Banach Spaces

In this paper, we extend the Moreau (Riesz) decomposition theorem from Hilbert spaces to Banach spaces. Criteria for a closed subspace to be (strongly) orthogonally complemented in a Banach space are given. We prove that every closed subspace of a Banach space X with dim X ≥ 3 (dim X ≤ 2) is strongly orthognally complemented if and only if the Banach space X is isometric to a Hilbert space (resp. strictly convex), which is complementary to the well-known result saying that every closed subspace of a Banach space X is topologically complemented if and only if the Banach space X is isomorphic to a Hilbert space.     

K-drop凸空间中的性质

为了阐明何为K 强光滑空间的对偶空间 ,本文定义了K drop凸空间并且讨论了该空间的一些性质。同时借助K 强光滑空间的一个等价定义 ,证明了K drop凸空间与K 强光滑空间是对偶空间。文章最后用单位圆的切片给出了K drop凸空间的一等价命题 ,进而建立了K drop凸空间与drop性之间的关系。     

On the degree and separability of nonconvexity and applications to optimization problems

We study qualitative indications for d.c. representations of closed sets in and functions on Hilbert spaces. The first indication is an index of nonconvexity which can be regarded as a measure for the degree of nonconvexity. We show that a closed set is weakly closed if this indication is finite. Using this result we can prove the solvability of nonconvex minimization problems. By duality a minimization problem on a feasible set in which this indication is low, can be reduced to a quasi-concave minimization over a convex set in a low-dimensional space. The second indication is the separability which can be incorporated in solving dual problems. Both the index of nonconvexity and the separability can be characteristics to “good” d.c. representations. For practical computation we present a notion of clouds which enables us to obtain a good d.c. representation for a class of nonconvex sets. Using a generalized Caratheodory’s theorem we present various applications of clouds.