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In this paper we investigate the fractional Poincar\'e inequality onunbounded domains. In the local case, Sandeep-Mancini showed that in the classof simply connected domains, Poincar\'e inequality holds if and only if thedomain does not allow balls of arbitrarily large radius (finite ballcondition). We prove that such a result can not be true in the`nonlocal/fractional' setting even if finite ball condition is replaced by arelated stronger condition. We further provide some sufficient criterions ondomains for fractional Poincar\'e inequality to hold. In the end, asymptoticbehaviour of all eigenvalues of fractional Dirichlet problems on longcylindrical domains is addressed.

Fractional Poincar\'e Inequality for Unbounded Domains with Finite Ball Condition: Counter Example