Strong Klee-And\^o Theorems through an Open Mapping Theorem for cone-valued multi-functions

TL;DR

This paper extends the classical Klee-And extsuperscript{o} Theorem to more general settings using an open mapping theorem for cone-valued multi-functions, providing stronger, continuous, and positively homogeneous decompositions in Banach spaces with cones.

Contribution

It introduces stronger versions of the Klee-And extsuperscript{o} Theorem for conormality and coadditivity, generalizes the theorem beyond ordered Banach spaces, and develops an open mapping theorem for cone-valued multi-functions.

Findings

Established stronger Klee-And extsuperscript{o} Theorems for conormality and coadditivity.

Generalized the theorem to Banach spaces with arbitrary collections of cones.

Proved an open mapping theorem for cone-valued multi-functions.

Abstract

A version of the classical Klee-And\^o 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 $\|x^{+}\|+\|x^{-}\|\leq\alpha\|x\|$. The conclusion of the Klee-And\^o Theorem is what is known as a conormality property. We prove stronger and somewhat more general versions of the Klee-And\^o 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-And\^o Theorem generalizes beyond ordered Banach spaces to Banach spaces endowed with arbitrary collections of cones. Proofs of our Klee-And\^o 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-And\^o 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 $x\in X$.