It is shown, using the Borwein–Preiss variational principle that for every continuous convex function f on a weakly compactly generated space X, every x0 ∈ X and every weakly compact convex symmetric set K such that spanK = X, there is a point of Gâteaux differentiability of f in x0 +K. This extends a Klee’s result for separable spaces. The well-known Mazur’s theorem says that a continuous conv...

We prove that Thompson's group F is not minimally almost convex with respect to any generating set which is a subset of the standard infinite generating set for F and which contains x1. We use this to show that F is not almost convex with respect to any generating set which is a subset of the standard infinite generating set, generalizing results in [HST].

We study the convexity property of the set QF of arbitrage-free prices of a multiperiod financial structure F . The set of arbitrage-free prices is shown to be a convex cone under conditions on the financial stucture F that hold in particular for short lived assets. Furthermore, we provide examples of equivalent financial structures F and F ′ such that QF is a convex cone, but QF ′ is neither c...

We first obtain some properties of a fundamentally nonexpansive self-mapping on a nonempty subset of a Banach space and next show that if the Banach space is having the Opial condition, then the fixed points set of such a mapping with the convex range is nonempty. In particular, we establish that if the Banach space is uniformly convex, and the range of such a mapping is bounded, closed and con...

and Applied Analysis 3 For given nonlinear operators T, g, h, we consider the problem of finding u ∈ H : h u ∈ K such that 〈 Tu, g v − h u 〉 ≥ 0, ∀v ∈ H : g v ∈ K, 2.1 which is called the extended general variational inequality. Noor 13–16 has shown that the minimum of a class of differentiable nonconvex functions on hg-convex set K in H can be characterized by extended general variational ineq...

we first obtain some properties of a fundamentally nonexpansive self-mapping on a nonempty subset of a banach space and next show that if the banach space is having the opial condition, then the fixed points set of such a mapping with the convex range is nonempty. in particular, we establish that if the banach space is uniformly convex, and the range of such a mapping is bounded, closed and con...

These notes will be rather informal in many places. For more precision, refer to Lectures on Polytopes by Günter Ziegler [Zie95], Convex Polytopes by Branko Grünbaum [Grü67], An Introduction to Convex Polytopes by Arne Brøndsted [Brø83], and Convex Polytopes and the Upper Bound Conjecture by Peter McMullen and Geoffrey Shephard [MS71]. See also the handbook papers [BL93, KK95] We will be studyi...

a definition of two jointly asymptotically nonexpansive mappings s and t on uniformly convex banach space e is studied to approximate common fixed points of two such mappings through weak and strong convergence of an ishikawa type iteration scheme generated by s and t on a bounded closed and convex subset of e. as a consequence of the notion of two jointly asymptotically nonexpansive maps, we c...

In the field of nonlinear programming (in continuous variables) convex analysis [22, 23] plays a pivotal role both in theory and in practice. An analogous theory for discrete optimization (nonlinear integer programming), called " discrete convex analysis " [18, 17], is developed for L-convex and M-convex functions by adapting the ideas in convex analysis and generalizing the results in matroid ...

This paper discusses computational complexity of conceptual successive convex relaxation methods proposed by Kojima and Tunn cel for approximating a convex relaxation of a compact subset F = fx 2 C 0 : p(x) 0 (8p() 2 P F)g of the n-dimensional Euclidean space R n. Here C 0 denotes a nonempty compact convex subset of R n , and P F a set of nitely or innnitely many quadratic functions. We evaluat...