Recession Cones of Nonconvex Sets and Increasing Functions

On Polar Cones and Differentiability in Reflexive Banach Spaces

Let $X$ be a  Banach  space, $Csubset X$  be  a  closed  convex  set  included  in  a well-based cone $K$, and also let $sigma_C$ be the support function which is defined on $C$. In this note, we first study the existence of a  bounded base for the cone $K$, then using the obtained results, we find some geometric conditions for the set  $C$,  so that ${mathop{rm int}}(mathrm{dom} sigma_C) neqem...

A Generalization of a Theorem of Arrow, Barankin and Blackwell to a Nonconvex Case

The paper presents a generalization of a known density theorem of Arrow, Barankin, and Blackwell for properly efficient points defined as support points of sets with respect to monotonically increasing sublinear functions. This result is shown to hold for nonconvex sets of a reflexive Banach space partially ordered by a Bishop–Phelps cone.

An effective optimization algorithm for locally nonconvex Lipschitz functions based on mollifier subgradients

Bishop-Phelps type Theorem for Normed Cones

In this paper the notion of  support points of convex sets  in  normed cones is introduced and it is shown that in a  continuous normed cone, under the appropriate conditions, the set of support points of a  bounded Scott-closed convex set is nonempty. We also present a Bishop-Phelps type Theorem for normed cones.

An Introduction to Convexity

Contents 1 The basic concepts 1 1.1 Is convexity useful? 1 1.2 Nonnegative vectors 4 1.3 Linear programming 5 1.4 Convex sets, cones and polyhedra 6 1.5 Linear algebra and affine sets 11 1.6 Exercises 14 2 Convex hulls and Carathéodory's theorem 17 2.1 Convex and nonnegative combinations 17 2.2 The convex hull 19 2.3 Affine independence and dimension 22 2.4 Convex sets and topology 24 2.5 Carat...