The Nonlinear Separation Theorem and a Representation Theorem for Bishop-Phelps Cones

The paper presents a theorem for representation a given cone as a Bishop-Phelps cone in normed spaces and studies interior and separation properties for Bishop-Phelps cones. The representation theorem is formulated in the form of a necessary and sufficient condition and provides relationship between the parameters determining the Bishop-Phelps cone. The necessity is given in reflexive Banach spaces. The representation theorem is used to establish the theorem on interior of the Bishop-Phelps cone representing a given cone, and the nonlinear separation theorem. It is shown that every Bishop-Phelps cone in finite dimensional space satisfies the separation property for the nonlinear separation theorem. The theorems on the representation, on the interior and on the separation property studied in this paper are comprehensively illustrated on examples in finite and infinite dimensional spaces.