Farthest points in weakly compact sets

LetS be a weakly compact subset of a Banach spaceB. We show that of all points inB which have farthest points inS contains a denseG ₅ ofB. Also, we give a necessary and sufficient condition for bounded closed convex sets to be the closed convex hull of their farthest points in reflexive Banach spaces.     

Farthest points of sets in uniformly convex banach spaces

LetS be a closed and bounded set in a uniformly convex Banach spaceX. It is shown that the set of all points inX which have a farthest point inS is dense. Letb(S) denote the set of all farthest points ofS, then a sufficient condition for to hold is thatX have the following property (I): Every closed and bounded convex set is the intersection of a family of closed balls.     

Local convexity and starshaped sets

Previously [7] we proved among other results that a closed connected set inE _n which has a unique point of local nonconvexity is starshaped. Here we characterize a fairly large class of plane sets whose points of local nonconvexity are so arranged that starshapedness follows. This theory determines as a special case the simple closed polygonal regions which are starshaped. In order to proceed simply we utilize the following notations and definitions. The preparation of this paper was supported in part by NSF Grant GP-1988.     

Compact convex sets and complex convexity

We construct a quasi-Banach space which cannot be given an equivalent plurisubharmonic quasi-norm, but such that it has a quotient by a one-dimensional space which is a Banach space. We then use this example to construct a compact convex set in a quasi-Banach space which cannot be affinely embedded into the spaceL ₀ of all measurable functions.     

Farthest points and cut loci on some degenerate convex surfaces

We study farthest points and cut loci on doubly covered convex polygons, and determine them explicitly on doubly covered n-dimensional simplices.     

关于Banach空间的强凸性

本文讨论强凸性、Ｌ－ｋＲ，ＬωＲ和（Ｇ）性质之间的关系，指出强凸性介于ＬωＲ和（Ｇ）性质之间，证明光滑的有（Ｇ）性质的Ｂａｎａｃｈ空间是强凸的，此外指出存在一个Ｂａｎａｃｈ空间Ｘ，它是ＬωＲ但对任意自然数ｋ，Ｘ不是Ｌ－ｋＲ．     

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.     

A difference between the sets of ordinary and strong density points on the plane

It is well known that the difference between the set of all ordinary density points and the set of all strong density points of an arbitrary measurable subset of the plane is a null set. It is of interest to check how large can a difference between sections of these sets be. Both measure and category cases are considered.     

Eventual convexity of chance constrained feasible sets

In decision-making problems where uncertainty plays a key role and decisions have to be taken prior to observing uncertainty, chance constraints are a strong modelling tool for defining safety of decisions. These constraints request that a random inequality system depending on a decision vector has to be satisfied with a high probability. The characteristics of the feasible set of such chance constraints depend on the constraint mapping of the random inequality system, the underlying law of uncertainty and the probability level. One characteristic of particular interest is convexity. Convexity can be shown under fairly general conditions on the underlying law of uncertainty and on the constraint mapping, regardless of the probability-level. In some situations, convexity can only be shown when the probability-level is high enough. This is defined as eventual convexity. In this paper, we will investigate further how eventual convexity can be assured for specially structured chance constraints involving Copulae. The Copulae have to exhibit generalized concavity properties. In particular, we will extend recent results and exhibit a clear link between the generalized concavity properties of the various mappings involved in the chance constraint for the result to hold. Various examples show the strength of the provided generalization.     

Polynomial convexity of some sets in Cn

Translated from Matematicheskie Zametki, Vol. 50, No. 5, pp. 81–89, November, 1991.     

Separation of two convex sets in convexity structures

A convexity structure satisfies the separation propertyS ₄ if any two disjoint convex sets extend to complementary half-spaces. This property is investigated for alignment spaces,n-ary convexities, and graphs. In particular, it is proven that

1. ann-ary convexity isS ₄ iff every pair of disjoint polytopes with at mostn vertices can be separated by complementary half spaces, and
2. an interval convexity isS ₄ iff it satisfies the analogue of the Pasch axiom of plane geometry.

A characterization of bipartite and weakly modular spaces withS ₄ convexity is given in terms of forbidden subgraphs.     

Non-Strebel points and variability sets

This paper studies the subset of the non-Strebel points in the universal Teichmuller space T. Let Z0 ∈ △be a fixed point. Then we prove that for every non-Strebel point h, there is a holomorphic curve γ : [0, 1]→ T with h as its initial point satisfying the following conditions.(1) The curve γ is on a sphere centered at the base-point of T, i.e. dT(id, γ(t)) = dT(id, h), (t ∈ [0, 1]).(2) For every t ∈ (0,1], the variability set Vγ(t)[Z0] of γ(t) has non-empty interior, i.e. Vγ(t) [Z0] ≠ .     

On some criteria of convexity for compact sets

We establish some criteria of convexity for compact sets in the Euclidean space. Analogs of these results are extended to complex and hypercomplex cases.     

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.