Swimming below icebergs

You are swimming close to an iceberg with a convex lower surface. You calculate at what slope you have to swim down so that, whatever the direction in which you swim, you can be sure that you will not collide with the iceberg. This limiting slope is intimately related to the existence of subtangents to the iceberg that satisfy various conditions. These considerations lead to generalizations of Rockafellar's Maximal Monotonicity Theorem, of which we give acomplete new proof. We also discuss related open problems on maximal monotonicity and subdifferentials, and generalizations of recent results on the existence of subtangents separating the epigraphs of proper convex lower semicontinuous functions from nonempty bounded closed convex sets, with some control over their slopes.     

凸距离空间内星形子集非扩张型映射不动点*

在本文中.我们给出包含一个集合的某种星形集的刻划及其性质.然后利用这些刻划和性质讨论凸距离空间的星形子集上非扩张型映射的不动点的存在问题,推广了丁协平、Beg和Azam的某些最近结果.最后还给出一个例子说明以上推广是本质上的推广.     

The kth lower bases of primitive non-powerful signed digraphs

In [J. Shao, L. You, H. Shan, Bound on the bases of irreducible generalized sign pattern matrices, Linear Algebra Appl. 427 (2007) 285-300], the authors extended the concept of the base from powerful sign pattern matrices to non-powerful irreducible sign pattern matrices. Recently, the kth local bases and the kth upper bases, which are generalizations of the bases, of primitive non-powerful signed digraphs were introduced. In this paper, we introduce a new parameter called the kth lower bases of primitive non-powerful signed digraphs and obtain some bounds for it. For some cases, the bounds we obtain are best possible and the extremal signed digraphs are characterized, respectively. Moreover, we show that there exist “gaps” in the kth lower bases set of primitive non-powerful signed digraphs.     

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.     

Erdős–Szekeres Theorem for Lines

According to the Erd?s–Szekeres theorem, for every n, a sufficiently large set of points in general position in the plane contains n in convex position. In this note we investigate the line version of this result, that is, we want to find n lines in convex position in a sufficiently large set of lines that are in general position. We prove almost matching upper and lower bounds for the minimum size of the set of lines in general position that always contains n in convex position. This is quite unexpected, since in the case of points, the best known bounds are very far from each other. We also establish the dual versions of many variants and generalizations of the Erd?s–Szekeres theorem.     

Local convergence of the method of multipliers for variational and optimization problems under the noncriticality assumption

We present an efficient algorithm to find an optimal fiber orientation in composite materials. Within a two-scale setting fiber orientation is regarded as a function in space on the macrolevel. The optimization problem is formulated within a function space setting which makes the imposition of smoothness requirements straightforward and allows for rather general convex objective functionals. We show the existence of a global optimum in the Sobolev space H ¹(Ω). The algorithm we use is a one level optimization algorithm which optimizes with respect to the fiber orientation directly. The costly solve of a big number of microlevel problems is avoided using coordinate transformation formulas. We use an adjoint-based gradient type algorithm, but generalizations to higher-order schemes are straightforward. The algorithm is tested for a prototypical numerical example and its behaviour with respect to mesh independence and dependence on the regularization parameter is studied.     

Convex polarities over ordered fields

We use tools and methods from real algebraic geometry (spaces of ultrafilters, elimination of quantifiers) to formulate a theory of convexity in K^N over an arbitrary ordered field. By defining certain ideal points (which can be viewed as generalizations of recession cones) we obtain a generalized notion of polar set. These satisfy a form of polar duality that applies to general convex sets and does not reduce to classical duality if K is the field of real numbers. As an application we give a partial classification of total orderings of Artinian local rings and two applications to ordinary convex geometry over the real numbers.     

Non-compact sets with convex sections,II

This paper is a continuation of previous work (Pacific J. Math. 119 (1985)) and contains two results on sets with convex sections involving four sets. As direct applications, we formulate generalizations of the von Neumann minimax theorem.     

Generalizations of Sherman’s inequality

The concept of majorization is a powerful and useful tool which arises frequently in many different areas of research. Together with the concept of Schur-convexity it gives an important characterization of convex functions. The well known Majorization theorem plays a very important role in majorization theory—it gives a relation between one-dimensional convex functions and n-dimensional Schur-convex functions. A more general result was obtained by S. Sherman. In this paper, we get generalizations of these results for n-convex functions using Taylor’s interpolating polynomial and the ?eby?ev functional. We apply the exponentially convex method in order to interpret our results in the form of exponentially, and in the special case logarithmically convex functions. The outcome is some new classes of two-parameter Cauchy-type means.     

A combinatorial proof of a theorem of Freund

In 1989, Robert W. Freund published an article about generalizations of the Sperner lemma for triangulations of n-dimensional polytopes, when the vertices of the triangulations are labeled with points of Rn. For y∈Rn, the generalizations ensure, under various conditions, that there is at least one simplex containing y in the convex hull of its labels. Moreover, if y is generic, there is generally a parity assertion, which states that there is actually an odd number of such simplices.For one of these generalizations, contrary to the others, neither a combinatorial proof, nor the parity assertion were established. Freund asked whether a corresponding parity assertion could be true and proved combinatorially.The aim of this paper is to give a positive answer, using a technique which can be applied successfully to prove several results of this type in a very simple way. We prove actually a more general version of this theorem. This more general version was published by van der Laan, Talman and Yang in 2001, who proved it in a non-combinatorial way, without the parity assertion.     

A measure of thickness for families of sets

A numerical characteristic is introduced for families of subsets of a given set S. It is shown that this characteristic may assume all values in the interval [0.1]. To this end suitable families of sets are constructed. Families for which this characteristic assumes the values zero are particularly important since they are related to a theorem on the existence of convex means proved by the second named author [5].     

On some generalization of “bang-bang” control

The theory of orientor fields is used to establish relations between systems with convex and nonconvex sets of admissible directions. It is pointed out that such systems have sets of quasitrajectories identical to each other. On the other hand, quasitrajectories are relevant generalizations of so-called sliding regimes, well-known in automatic control. Control functions of “bang-bang” type appear to be, in turn, some cases of controls generated by tender kernels of control domains of systems to be considered. This might be applied for example to systems described by partial differential equations or to systems with state vectors in lⁿ spaces. The possibilities of further generalizations concerning optimality conditions are indicated.     

A generalized distance and enhanced Ekeland’s variational principle for vector functions

We propose a definition of lower closed transitive relations and prove the existence of minimal elements for such a relation. This result is shown to contain probably a large part of existing versions of Ekeland’s variational principle (EVP). We introduce the notion of a weak τ-function p as a generalized distance and use it together with the above result on minimal elements to establish enhanced EVP for various settings, under relaxed lower semicontinuity assumptions. These principles conclude the existence not only of p-strict minimizers of p-perturbations of the considered vector function, but also p-sharp and p-strong minimizers. Our results are proved to be stronger than the classical EVP and many generalizations in the literature, even in the usual one-dimensional case, by numerous corollaries and examples. We include equivalent formulations of our enhanced EVP as well.     

Construction of a linear Pfaff system with <Emphasis Type="Italic">m</Emphasis>-dimensional time and with given characteristic set and lower characteristic set

For any positive integers n ≥ 1 and m ≥ 2, we give a constructive proof of the existence of linear n-dimensional Pfaff systems with m-dimensional time and with infinitely differentiable coefficient matrices such that the characteristic and lower characteristic sets of these systems are given sets that are the graphs of a concave continuous function and a convex continuous function, respectively, defined and monotone decreasing on simply connected closed bounded convex domains of the space ?^m?1.     

The concept of approximative compactness and alternate versions of it

We study various characteristics and generalizations of approximative compact and approximative weak compact sets. We generalize a result of Asplund concerning sets whose intersection with each halfspace is an existence set. In particular, in smooth Efimov-Stechkin spaces, such a set, if it is a Chebyshev set, must be convex.     

Applications of Michael's selection theorems to fixed point theory

Applying some of Ernest Michael's selection theorems, from recent fixed point theorems on u.s.c. multimaps, we deduce generalizations of the classical Bolzano theorem, several fixed point theorems on multimaps defined on almost convex sets, almost fixed point theorems, coincidence theorems, and collectively fixed point theorems. These results are related mainly to Michael maps, that is, l.s.c. multimaps having nonempty closed convex values.     

Completions of convex families of convex bodies

The paper discusses the existence of a continuous extension of functions that are defined on subsets of ? ⁿ and whose values are convex bodies in ? ⁿ . This problem arose in convex geometry in connection with the notion, recently introduced in algebraic geometry, of convex Newton-Okounkov bodies.     

An equilibrium version of set-valued Ekeland variational principle and its applications to set-valued vector equilibrium problems

By using Gerstewitz functions, we establish a new equilibrium version of Ekeland variational principle, which improves the related results by weakening both the lower boundedness and the lower semi-continuity of the ob jective bimaps. Applying the new version of Ekeland principle, we obtain some existence theorems on solutions for set-valued vector equilibrium problems, where the most used assumption on compactness of domains is weakened. In the setting of complete metric spaces(Z,d), we present an existence result of solutions for set-valued vector equilibrium problems, which only requires that the domain XZ is countably compact in any Hausdorff topology weaker than that induced by d. When(Z, d) is a Féchet space(i.e., a complete metrizable locally convex space), our existence result only requires that the domain XZ is weakly compact. Furthermore, in the setting of non-compact domains, we deduce several existence theorems on solutions for set-valued vector equilibrium problems,which extend and improve the related known results.     

On (n,m)-Convex Sets

We investigate the class of generalized convex sets on Grassmann manifolds, which includes known generalizations of convex sets for Euclidean spaces. We extend duality theorems (of polarity type) to a broad class of subsets of the Euclidean space. We establish that the invariance of a mapping on generalized convex sets is equivalent to its affinity.     

A Proof of the Convexity of the Free Boundary for Porous Flow Through a Rectangular Dam Using the Maximum Principle

We consider the steady two-dimensional flow under gravity ofwater from one reservoir (on the left) to a lower reservoir(on the right) through a porous rectangular isotropic homogeneousdam with impervious bottom. Because of gravity the water doesnot flow through the entire dam and the dam is dry near itsupper right corner. The interface separating the dry and wetregions of the dam is a free boundary. Recently, Friedman &Jensen (1977) have proved that the free boundary is convex.We give a different proof which uses only the maximum principleand its generalizations.