This is an announcement for the paper “Porosity Results for Sets of Strict Contractions on Geodesic Metric Spaces” by Christian Bargetz, Michael Dymond and Simeon Reich.

Abstract: We consider a large class of geodesic metric spaces, including Banach spaces, hyperbolic spaces and geodesic $CAT(\kappa)$-spaces, and investigate the space of nonexpansive mappings on either a convex or a star-shaped subset in these settings. We prove that the strict contractions form a negligible subset of this space in the sense that they form a $\sigma$-porous subset. For separable metric spaces we show that a generic nonexpansive mapping has Lipschitz constant one at typical points of its domain. These results contain the case of nonexpansive self-mappings and the case of nonexpansive set-valued mappings as particular cases.

The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1602.05230