In this paper, we consider pursuit–evasion and probabilistic consequences of some geometric notions for bounded and suitably regular domains in Euclidean space that are CAT(κ) for some κ>0 . These geometric notions are useful for analysing the related problems of (a) existence/ non-existence of successful evasion strategies for the Man in Lion and Man problems, and (b) existence/non-existence of shy couplings for reflected Brownian motions. They involve properties of rubber bands and the extent to which a loop in the domain in question can be deformed to a point without, in between, increasing its loop length. The existence of a stable rubber band will imply the existence of a successful evasion strategy but, if all loops in the domain are well-contractible, then no successful evasion strategy will exist and there can be no co-adapted shy coupling. For example, there can be no shy couplings in bounded and suitably regular star-shaped domains and so, in this setting, any two reflected Brownian motions must almost surely make arbitrarily close encounters as t→∞.
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