We use variational methods to provide a concise development of a number of basic results in convex and functional analysis. This illuminates the parallels between convex analysis and smooth subdifferential theory. 1. The purpose of this note is to give a concise and explicit account of the following folklore: several fundamental theorems in convex analysis such as the sandwich theorem and the F...

Let F be a family of translates of a fixed convex set M in Rn. Let (F) and (F) denote the transversal number and the independence number of F, respectively. We show that (F) (F) 8 (F) − 5 for n = 2 and (F) 2n−1nn (F) for n 3. Furthermore, if M is centrally symmetric convex body in the plane, then (F) (F) 6 (F)− 3. © 2006 Elsevier B.V. All rights reserved.

Let $G=(V,E)$ be a simple graph. A set $Dsubseteq V$ is adominating set of $G$ if every vertex in $Vsetminus D$ has atleast one neighbor in $D$. The distance $d_G(u,v)$ between twovertices $u$ and $v$ is the length of a shortest $(u,v)$-path in$G$. An $(u,v)$-path of length $d_G(u,v)$ is called an$(u,v)$-geodesic. A set $Xsubseteq V$ is convex in $G$ ifvertices from all $(a, b)$-geodesics belon...

Let F be a family of sets in Rd (always d≥2). A set M⊂Rd is called F-convex, if for any pair distinct points x,y∈M, there F∈F, such that x,y∈F and F⊂M. thin right triangle the boundary non-degenerate R2. The aim this paper to introduce begin investigating convexity short trt-convexity, which obtained when all triangles. We investigate trt-convexity unbounded sets, convex surfaces planar geometr...

We study the minimization problem f (x) → min, x ∈ C, where f belongs to a complete metric space of convex functions and the set C is a countable intersection of a decreasing sequence of closed convex sets Ci in a reflexive Banach space. Let be the set of all f ∈ for which the solutions of the minimization problem over the set Ci converge strongly as i→∞ to the solution over the set C. In our r...

Let Z andX be Banach spaces, U ⊂ Z be an open convex set and f : U → X be a mapping. We say that f is a d.c. mapping if there exists a continuous convex function h on U such that y ◦ f + h is a continuous convex function for each y ∈ Y , ‖y∗‖ = 1. We say that f : U → X is locally d.c. if for each x ∈ U there exists open convex U ′ such that x ∈ U ′ ⊂ U and f |U ′ is d.c. This notion of d.c. map...

The local quadratic convergence of the Gauss-Newton method for convex composite optimization f = h ◦ F is established for any convex function h with the minima set C, extending Burke and Ferris’ results in the case when C is a set of weak sharp minima for h.

Let % be a preference relation on a convex set F . Necessary and sufficient conditions are given that guarantee the existence of a set {ul} of affine utility functions on F such that % is represented by U ( f ) = ul ( f ) if f ∈ Fl; where each Fl is a convex subset of F . The interpretation is simple: facing a “non-homogeneous” set F of alternatives, a decision maker splits it into “homogeneous...

Alexander [1] was the first to introduce certain subclasses of univalent functions examining the geometric properties of the image f(D) of D under f . The convex functions are those that map D onto a convex set. A function w = f(z) is said to be starlike if, together with any of its points w, the image f(D) contains the entire segment {tw : 0 ≤ t ≤ 1}. Thus we introduce the denotations S = {f ∈...

In 1906 Jensen founded the theory of convex functions. This enabled him to prove a considerable extension of the AM-GM inequality. Recall that a subset D of a real vector space is called convex if every convex linear combination of a pair of points of D is in D. Equivalently, if x, y ∈ D, then tx+ (1− t)y ∈ D for every t ∈ [0, 1]. Given a convex set D, a function f : D → R is called convex if f...