Porosity results for sets of strict contractions on geodesic metric spaces

Christian Bargetz, Michael Dymond, Simeon Reich

DOI: http://dx.doi.org/10.12775/TMNA.2017.013

Abstract


We consider a large class of geodesic metric spaces, including Banach spaces, hyperbolic spaces and geodesic $\mathrm{CAT}(\kappa)$-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 certain separable and complete 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.

Keywords


Banach space; hyperbolic space; metric space; nonexpansive mapping; porous set; set-valued mapping; star-shaped set; strict contraction

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