Some criteria for the minimality of pairs of compact convex sets

Some criteria for minimality for a pair of compact convex subsets of a real locally convex vector space are proved. Moreover several examples are given.     

On Partially Ordered Semigroups and an Abstract Set-Difference

In this paper we prove the independence of a system of five axioms (S1)–(S5), which was proposed in the book of Pallaschke and Urbański (Pairs of Compact Convex Sets, vol. 548, Kluwer Academic Publishers, Dordrecht, 2002) for partially ordered commutative semigroups. These five axioms (S1)–(S5) are stated in the introduction below. A partially ordered commutative semigroup satisfying these axioms is called a F-semigroup. By the use of a further axiom (S6) we define an abstract difference for the elements of a F-semigroup and prove some basic properties. The most interesting example of a F-semigroup are the nonempty compact convex sets of a topological vector space endowed with the Minkowski sum as operation and the inclusion as partial order. In Section 4 we apply the abstract difference to the problem of minimality of convex fractions. Dedicated to Boris Mordukhovich in honour of his 60th birthday.     

Representation of the Clarke Generalized Jacobian via the Quasidifferential

Two differences of convex compact sets in ^{m× n} are proposed. In the light of these differences, representations of the Clarke generalized Jacobian and the B-differential via the quasidifferential are developed for a certain class of functions. These representations can be used to calculate the Clarke generalized Jacobian and the B-differential via the quasidifferential.     

Differences of Polyhedra in Matrix Space and Their Applications to Nonsmooth Analysis

Formulas of the differences of polyhedra in matrix space are proposed. Based on these formulas, the differences of polyhedra can be calculated by solving systems of linear inequalities. A modified algorithm for calculating one element of the differences is presented also. The motivation for this work is to compute the Clarke generalized Jacobian, the B-differential, and one of their elements via the quasidifferential. Applications to Newton methods for solving nonsmooth equations are discussed.This project was sponsored by the Shanghai Education Committee, Grant 04EA01, by the Education Ministry of China, and by the Shanghai Government, Grant T0502. The author thanks two anonymous referees and Professor F. Giannessi for valuable suggestions and comments.     

Descriptive properties of the set of exposed points of compact convex sets in

We construct a compact convex subset of such that the set of its exposed points is not the intersection of an set and a set. The existence of such a set answers a question posed by G. Choquet, H.H. Corson and V.L. Klee.

     

诱导极限中的弱紧集

本文研究了诱导极限中的弱集与紧集，我们给出了诱导极限ｉｎｄｉｍ Ｅｎ中每个弱紧集含于且弱紧于某Ｅｎ的一些条件。     

Differences of Convex Compact Sets in the Space of Directed Sets. Part I: The Space of Directed Sets

A normed and partially ordered vector space of so-called directed sets is constructed, in which the convex cone of all nonempty convex compact sets in R ⁿ is embedded by a positively linear, order preserving and isometric embedding (with respect to a new metric stronger than the Hausdorff metric and equivalent to the Demyanov one). This space is a Banach and a Riesz space for all dimensions and a Banach lattice for n=1. The directed sets in R ⁿ are parametrized by normal directions and defined recursively with respect to the dimension n by the help of a support function and directed supporting faces of lower dimension prescribing the boundary. The operations (addition, subtraction, scalar multiplication) are defined by acting separately on the support function and recursively on the directed supporting faces. Generalized intervals introduced by Kaucher form the basis of this recursive approach. Visualizations of directed sets will be presented in the second part of the paper.     

一类局部凸空间及其映照特征

称局部凸空间（E,(?)0）为WCM空间若对于任何弱于(?)0的局部凸拓扑(?),（E,(?)）与（E,(?)0）具相同的弱紧圆凸集.本文研究了WCM空间的存在性及其与其他类型局部凸空间之间的关系,还给出了WCM空间的一种映照特征.     

Relatively Compact Sets on Abstract Wiener Space

In this note, we obtain a sufficient and necessary condition for a set in an abstract Winner space （X, H, μ） to be relatively compact in L^2（X, μ）. Meanwhile, we give a sufficient condition for relative compactness in L^P（X, μ） for p〉1. We also provide an example of Da Prato-Malliavin Nualart to show the result.     

On Shapiro's results in quasidifferential calculus

Several necessary minimum conditions for quasidifferentiable programming problems are discussed. Some statements of Shapiro (1986) are shown to be incorrect.     

On Decompositions of Compact Convex Sets

The present paper generalizes M. Edelstein's theorem on the indecomposability of compact convex sets in locally convex linear topological spaces to spherical and hyperbolic geometry. Moreover, the indecomposability of compact intervals in EU¹ w.r.t. homeomorphisms of EU¹ onto itself is shown.     

Directional derivative estimates for the optimal value function of a quasidifferentiable programming problem

This paper is concerned with the optimal value function arising in the primal decomposition of a quasidifferentiable programming problem. In particular, estimates for the upper Dini directional derivative of this function are derived. They involve certain Lagrange multipliers occurring in the necessary minimum conditions to the lower level problems. This study generalizes some previously published results on this subject.     

两个凸多面体的差及几种次微分之间的关系

给出两种两个凸多面体差的表达式,利用这些表达式,可以具体计算这两种凸多面体的差,做为应用讨论了利用拟微分计算Penot微分和Clarke广义梯度,特别讨论了一类非光滑函数,极大值函数的光滑复合。     

On inclusion and summands of bounded closed convex sets

Summary This article introduces coset extensions and group coextensions of S-sets.     

A Bipartite Analogue of Dilworth’s Theorem

Let m(n) be the maximum integer such that every partially ordered set P with n elements contains two disjoint subsets A and B, each with cardinality m(n), such that either every element of A is greater than every element of B or every element of A is incomparable with every element of B. We prove that . Moreover, for fixed ε ∈ (0,1) and n sufficiently large, we construct a partially ordered set P with n elements such that no element of P is comparable with other elements of P and for every two disjoint subsets A and B of P each with cardinality at least , there is an element of A that is comparable with an element of B.     

Operations on approximatively compact sets

In the paper, the problem of preserving the property of approximative compactness under diverse operations is considered. In an arbitrary uniformly convex separable space, we construct an example of two approximatively compact sets whose intersection is not approximatively compact. An example of two linear approximatively compact sets for which the closure of their algebraic sum is not approximatively compact is constructed. In an arbitrary Banach space, we construct two nonlinear approximatively compact sets whose algebraic sum is closed but not approximatively compact. We also prove that any uniformly closed Banach space contains an approximatively compact cavity.     

数值函数关于集值测度的积分的极限定理

本文给出了实值函数关于集值测度的积分序列的极限定理．     

A minimal set-valued strong derivative for vector-valued Lipschitz functions

A set-valued derivative for a function at a point is a set of linear transformations whichapproximates the function near the point. This is stated precisely, and it is shown that, in general, there is not a unique minimal set-valued derivative for functions in the family of closed convex sets of linear transformations. For Lipschitz functions, a construction is given for a specific set-valued derivative, which reduces to the usual derivative when the function is strongly differentiable, and which is shown to be the unique minimal set-valued derivative within a certain subfamily of the family of closed convex sets of linear transformations. It is shown that this constructed set may be larger than Clarke's and Pourciau's set-valued derivatives, but that no irregularity is introduced.The author would like to thank Professor H. Halkin for numerous discussions of the material contained here.