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arXiv · 1602.05230

Porosity Results for Sets of Strict Contractions on Geodesic Metric Spaces

Christian Bargetz·Michael Dymond·Simeon Reich
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Abstract

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

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Christian Bargetz, Michael Dymond, Simeon Reich. 2017-02-12. Porosity Results for Sets of Strict Contractions on Geodesic Metric Spaces. https://doi.org/10.12775/tmna.2017.013

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