This is an announcement for the paper “A Pointwise Lipschitz Selection Theorem $” by Miek Messerschmidthttps://arxiv.org/find/math/1/au:+Messerschmidt_M/0/1/0/all/0/1.

Abstract: We prove that any correspondence (multi-function) mapping a metric space into a Banach space that satisfies a certain pointwise Lipschitz condition, always has a continuous selection that is pointwise Lipschitz on a dense set of its domain. We apply our selection theorem to demonstrate a slight improvement to a well-known version of the classical Bartle-Graves Theorem: Any continuous linear surjection between infinite dimensional Banach spaces has a positively homogeneous continuous right inverse that is pointwise Lipschitz on a dense meager set of its domain. An example devised by Aharoni and Lindenstrauss shows that our pointwise Lipschitz selection theorem is in some sense optimal: It is impossible to improve our pointwise Lipschitz selection theorem to one that yields a selection that is pointwise Lipschitz on the whole of its domain in general.

The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.08435