Criteria for convexity in Banach spaces

In this paper two convexity criteria are proven. The first one characterizes compact convex sets in a locally convex space and extends a previous result by G. Aumann, while the second one characterizes closed bounded convex sets with the Radon-Nikodým property in a Banach space.

     

Near convexity,near smoothness and approximative compactness of half spaces in Banach spaces

The authors discuss the dual relation of nearly very convexity and property WS. By two kinds of near convexities and two kinds of near smoothness, the authors prove a series of characterizations such that every half-space in Banach space X and every weak* half-space in the dual space X* are approximatively weakly compact and approximatively compact. They show a sufficient condition such that a Banach space X is a Asplund space. Using upper semi-continuity of duality mapping, the authors also give two characterizations of property WS and property S.     

The integral convexity of sets and functionals in Banach spaces

By way of the Bochner integral of vector-valued functions, the integral convexity of sets and functionals and the concept of integral extreme points of sets are introduced in Banach spaces. The relations between integral convexity and convexity are mainly discussed, two integral extreme points theorems and their applications are obtained at last.     

Orlicz spaces with convexity or concavity constant one

We characterize the classes of function and sequence Orlicz spaces that satisfy upper or lower p-estimate with constant one. We also present a characterization of p-convexity or p-concavity with constant one in Orlicz spaces.     

Banach spaces in which a theorem of orlicz is not true

Let the Banach space X be such that for every numerical sequencet _n ↘0 there exists in X an unconditionally convergent series σx_n, the terms of which are subject to the condition ∥x_n∥=t_n (n=1,2,...). Then $$\mathop {sup}\limits_n \mathop {inf}\limits_{X_n } d(X_n ,l_\infty ^n )< \infty ,$$ where X_n ranges over all the n-dimensional subspaces of X.     

On the convexity of the weakly compact Chebyshev sets in Banach spaces

A sufficient condition for a Banach spaceX is given so that every weakly compact Chebyshev subset ofX is convex. For this purpose a class broader than that of smooth Banach spaces is defined. In this way a former result of A. Brøndsted and A. L. Brown is partially extended in every finite dimensional normed linear space and a known result in Hilbert spaces is proved in a different way.     

On the convexity of C 1 surfaces associated with some quadrilateral finite elements

We analyse the convexity property of two classical finite elements and of the associated piecewise polynomial C ¹ surfaces. The first one is the piecewise cubic quadrilateral of Fraeijs de Veubeke. The second one is a piecewise quadratic rectangle introduced by Sibson and Thomson and generalized by Sablonnière and Zedek. In both cases, we first study the local problem and then we extend our results to the associated C ¹ surfaces. This revised version was published online in June 2006 with corrections to the Cover Date.     

Risk measures with comonotonic subadditivity or convexity on product spaces

In this paper, by an axiomatic approach, we propose the concepts of comonotonic subadditivity and comonotonic convex risk measures for portfolios, which are extensions of the ones introduced by Song and Yan(2006)Representation results for these new introduced risk measures for portfolios are given in terms of Choquet integralsLinks of these newly introduced risk measures to multi-period comonotonic risk measures are representedFinally, applications of the newly introduced comonotonic coherent risk measures to capital allocations are provided.     

Minimum selections and fixed points of set-valued operators in Banach spaces with some uniform convexity

A new continuity theorem of minimum selection is presented for a continuous set-valued operator from a topological space into a Banach space with some uniform convexity. As applications, some problems concerning minimum right inverses for linear operators and minimum fixed points for condensing set-valued nonlinear operators are discussed. Also, the existence of minimum solutions for an integral inclusion is proved.     

Simplices with edges of equal length in finite dimensional Banach spaces

Natural generalizations of planar results on volume ratios with respect to convex bodies in finite dimensional Banach spaces are given. The underlying paper contains inequalities for the volume of simplices with edges of equal length in a Banach space.Dedicated to Professor Helmut Mäurer on the occasion of his 60th birthday     

Uniformly smooth renorming of Banach spaces with modulus of convexity of power type 2

An upper bound q(c) for the best, under equivalent renorming, possible power type of the modulus of smoothness of a Banach space with modulus of convexity satisfying δX(ε)?cε², is found. The estimate is asymptotically sharp and is expressed in terms of linear fractional function q(c).     

Banach spaces in which Dunford-Pettis sets are relatively compact

Work performed under the auspices of G.N.A.F.A. of C.N.R. and partially supported by M.U.R.S.T. of Italy (40%)     

Relations between some constants associated with finite dimensional Banach spaces

We define the asymmetry constants(E) of a Banach spaceE, and show examples of finite-dimensional spaces with “large” asymmetry constants. IfE isn-dimensional,λ(E)17its projection constant and π ₁(I _E ) the absolutely summing norm of the identity operatorI _E , thenn≦λ(E)π₁(I _E )≤n(s(E))². Similar equations linking thep-absolutely summing and the nuclear norms ofI _E are established. We also obtain estimates on these norms, for example π₂(I _E )=√n. The contribution of this author is a part of a Ph.D. Thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor J. Lindenstrauss whose guidance and valuable suggestions are gratefully acknowledged.     

Banach ideals of operators with applications to the finite dimensional structure of Banach spaces

We present a few applications of the theory of Banach ideals of operators. In particular, we give operator characterizations of the ℒ _p spaces, compute the relative projection constant of isometric embeddings of Hilbert spaces inL _p -spaces, and show that Π₁ (E, F), the space of absolutely summing operators, is reflexive ifE andF are reflexive andE has the approximation property. Research supported by NSF-GP-34193 Research supported by NSF-Science Development Grant