New integral representations of nth order convex functions

In this paper we give an integral representation of an n-convex function f in general case without additional assumptions on function f. We prove that any n-convex function can be represented as a sum of two (n+1)-times monotone functions and a polynomial of degree at most n. We obtain a decomposition of n-Wright-convex functions which generalizes and complements results of Maksa and Páles (2009) [13]. We define and study relative n-convexity of n-convex functions. We introduce a measure of n-convexity of f. We give a characterization of relative n-convexity in terms of this measure, as well as in terms of nth order distributional derivatives and Radon-Nikodym derivatives. We define, study and give a characterization of strong n-convexity of an n-convex function f in terms of its derivative f(n+1)(x) (which exists a.e.) without additional assumptions on differentiability of f. We prove that for any two n-convex functions f and g, such that f is n-convex with respect to g, the function g is the support for the function f in the sense introduced by W?sowicz (2007) [29], up to polynomial of degree at most n.     

Strongly hyperbolically convex functions

Let C(w₁,w₂,w₃) denote the circle in through w₁,w₂,w₃ and let denote one of the two arcs between w₁,w₂ belonging to C(w₁,w₂,w₃). We prove that a domain Ω in the Riemann sphere, with no antipodal points, is spherically convex if and only if for any w₁,w₂,w₃∈Ω, with w₁≠w₂, the arc of the circle which does not contain lies in Ω. Based on this characterization we call a domain G in the unit disk D, strongly hyperbolically convex if for any w₁,w₂,w₃∈G, with w₁≠w₂, the arc in D of the circle is also contained in G. A number of results on conformal maps onto strongly hyperbolically convex domains are obtained.     

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

     

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.     

The uniform convexity of the Edmunds-Triebel logarithmic spaces

We show how uniform convexity can be preserved in the logarithmic spaces A_θ(logA)_b,p. Estimates are given for the moduli of convexity of A_θ(logA)_b,p in terms of the moduli of A₀ and A₁, when one or both of them are uniformly convex.     

Canonical and monophonic convexities in hypergraphs

Known properties of “canonical connections” from database theory and of “closed sets” from statistics implicitly define a hypergraph convexity, here called canonical convexity (c-convexity), and provide an efficient algorithm to compute c-convex hulls. We characterize the class of hypergraphs in which c-convexity enjoys the Minkowski-Krein-Milman property. Moreover, we compare c-convexity with the natural extension to hypergraphs of monophonic convexity (or m-convexity), and prove that: (1) m-convexity is coarser than c-convexity, (2) m-convexity and c-convexity are equivalent in conformal hypergraphs, and (3) m-convex hulls can be computed in the same efficient way as c-convex hulls.     

Characterizations of universal finite representability and b-convexity of Banach spaces via ball coverings

By a ball-covering B of a Banach space X, wemean that B is a collection of open (or closed) balls off the origin whose union contains the unit sphere of X; and X is said to have the ball-covering property provided it admits a ball-covering of countably many balls. This paper shows that universal finite representability and B-convexity of X can be characterized by properties of ball-coverings of its finite dimensional subspaces.     

Noncommutative convexity arises from linear matrix inequalities

This paper concerns polynomials in g noncommutative variables x=(x₁,…,xg), inverses of such polynomials, and more generally noncommutative “rational expressions” with real coefficients which are formally symmetric and “analytic near 0.” The focus is on rational expressions r=r(x) which are “matrix convex” near 0; i.e., those rational expressions r for which there is an ?>0 such that if X=(X₁,…,Xg) is a g-tuple of n×n symmetric matrices satisfying

     

Some inequalities concerning the James constant in Banach spaces

Let A₂(X) be the constant introduced by Baronti, Casini and Papini. This paper discusses the constant A₂(X) and states an estimate in terms of the James constant. The estimate enables us to improve an inequality between the James and von Neumann-Jordan constants.     

Approximate convexity of Takagi type functions

Given a bounded function Φ:R→R, we define the Takagi type function TΦ:R→R by

     

On a class of Köthe sequence spaces with normal structure

In this note, we investigate the generalized modulus of convexity δ ( λ ) and the generalized modulus smoothness ρ ( λ ) . We obtain some estimates of these moduli for X = lp . We obtain inequalities between WCS coefficient of a K¨othe sequence space X and δ ( λ ) X . We show that, for a wide class of K¨othe sequence spaces X, if for some ε∈ (0, 9 10 ] holds δ X (ε) > 1 3 1 √ 3 2 ε, then X has normal structure.     

On some relative convexities

We study two types of relative convexities of convex functions f and g. We say that f is convex relative to g   in the sense of Palmer (2002, 2003), if f=h(g)$f=h(g)$, where h   is strictly increasing and convex, and denote it by f_?(1)g$f{?}_{(1)}g$. Similarly, if f is convex relative to g   in the sense studied in Rajba (2011), that is if the function f−g$f-g$ is convex then we denote it by f_?(2)g$f{?}_{(2)}g$. The relative convexity relation _?(2)${?}_{(2)}$ of a function f   with respect to the function g(x)=cx²$g(x)=c{x}^2$ means the strong convexity of f. We analyze the relationships between these two types of relative convexities. We characterize them in terms of right derivatives of functions f and g, as well as in terms of distributional derivatives, without any additional assumptions of twice differentiability. We also obtain some probabilistic characterizations. We give a generalization of strong convexity of functions and obtain some Jensen-type inequalities.     

Extensions of convex and semiconvex functions and intervally thin sets

We call A⊂RNintervally thin if for all x,y∈RN and ε>0 there exist x^′∈B(x,ε), y^′∈B(y,ε) such that [x^′,y^′]∩A=∅. Closed intervally thin sets behave like sets with measure zero (for example such a set cannot “disconnect” an open connected set). Let us also mention that if the (N−1)-dimensional Hausdorff measure of A is zero, then A is intervally thin. A function f is preconvex if it is convex on every convex subset of its domain. The consequence of our main theorem is the following: Let U be an open subset ofRNand let A be a closed intervally thin subset of U. Then every preconvex functioncan be uniquely extended (with preservation of preconvexity) onto U. In fact we show that a more general version of this result holds for semiconvex functions.     

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.     

Some remarks on the Minty vector variational principle

In scalar optimization it is well known that a solution of a Minty variational inequality of differential type is a solution of the related optimization problem. This relation is known as “Minty variational principle.” In the vector case, the links between Minty variational inequalities and vector optimization problems were investigated in [F. Giannessi, On Minty variational principle, in: New Trends in Mathematical Programming, Kluwer Academic, Dordrecht, 1997, pp. 93-99] and subsequently in [X.M. Yang, X.Q. Yang, K.L. Teo, Some remarks on the Minty vector variational inequality, J. Optim. Theory Appl. 121 (2004) 193-201]. In these papers, in the particular case of a differentiable objective function f taking values in Rm and a Pareto ordering cone, it has been shown that the vector Minty variational principle holds for pseudoconvex functions. In this paper we extend such results to the case of an arbitrary ordering cone and a nondifferentiable objective function, distinguishing two different kinds of solutions of a vector optimization problem, namely ideal (or absolute) efficient points and weakly efficient points. Further, we point out that in the vector case, the Minty variational principle cannot be extended to quasiconvex functions.     

On convex decompositions of a planar point set

Let P be a planar point set in general position. Neumann-Lara et al. showed that there is a convex decomposition of P with at most elements. In this paper, we improve this upper bound to .     

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.     

Sufficiency and duality in nondifferentiable multiobjective programming involving generalized type I functions

Recently, Hachimi and Aghezzaf defined generalized (F,α,ρ,d)-type I functions, a new class of functions that unifies several concepts of generalized type I functions. In this paper, the generalized (F,α,ρ,d)-type I functions are extended to nondifferentiable functions. By utilizing the new concepts, we obtain several sufficient optimality conditions and prove mixed type and Mond-Weir type duality results for the nondifferentiable multiobjective programming problem.