Strong Klee-Andô Theorems through an Open Mapping Theorem for cone-valued multi-functions

Abstract

A version of the classical Klee-Andô Theorem states the following: For every Banach space $X$, ordered by a closed generating cone $C\subseteq X$, there exists some $\alpha >0$ so that, for every $x\in X$, there exist $x^{\pm }\in C$ so that $x=x^{+}-x^{-}$ and $\Vert x^{+}\Vert +\Vert x^{-}\Vert \le \alpha \Vert x\Vert$. The conclusion of the Klee-Andô Theorem is what is known as a conormality property. We prove stronger and somewhat more general versions of the Klee-Andô Theorem for both conormality and coadditivity (a property that is intimately related to conormality). A corollary to our result shows that the functions $x\mapsto x^{\pm }$, as above, may be chosen to be bounded, continuous, and positively homogeneous, with a similar conclusion yielded for coadditivity. Furthermore, we show that the Klee-Andô Theorem generalizes beyond ordered Banach spaces to Banach spaces endowed with arbitrary collections of cones. Proofs of our Klee-Andô Theorems are achieved through an Open Mapping Theorem for cone-valued multi-functions/correspondences. We very briefly discuss a potential further strengthening of The Klee-Andô Theorem beyond what is proven in this paper, and motivate a conjecture that there exists a Banach space $X$, ordered by a closed generating cone $C\subseteq X$, for which there exist no Lipschitz functions $(\cdot {)}^{\pm }:X\to C$ satisfying $x=x^{+}-x^{-}$ for all $x\in X$.

Major rewrite. Large parts were removed which a referee pointed out can be proven through much easier methods

References in corpus (3)

• Normality of spaces of operators and quasi-lattices
• A strong open mapping theorem for surjections from cones onto Banach spaces
• Geometric duality theory of cones in dual pairs of vector spaces