On the Lipschitz Decomposition Problem in Ordered Banach Spaces and Its Connections to Other Branches of Mathematics

Abstract

Consider the following still open problem: For any Banach space X, ordered by a closed generating cone C ⊆ X, does there always exist Lipschitz functions (•) ± : X → C satisfying x = x + − x − for every x ∈ X?We discuss the connections of this problem to a large number of other branches of mathematics: set-valued analysis, selection theorems, the nonlinear geometry of Banach spaces, Ramsey theory, Lipschitz function spaces, duality theory, and tensor products of Banach spaces. We give equivalent reformulations of the problem, and, through known examples, provide circumstantial evidence that the above question could be answered in the negative.

A family of quotient maps of $\ell^\infty$ that do not admit uniformly continuous right inverses

“…If the quotient X/E is separable and the quotient map q : X → X/E has a Lipschitz right inverse, then q also has a continuous linear right inverse, i.e., the subspace E is complemented. To the author's knowledge (and apparently that of the MathOverflow community [9]), to date, there are only two examples of Banach space quotient maps that do not admit uniformly continuous or Lipschitz right inverses known. They are:…”