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In this paper, we give an optimal inequality relating the relative Yamabeinvariant of a certain compactification of a conformally compactPoin-car{\'e}-Einstein manifold with the Yamabe invariant of its boundary atinfinity. As an application, we obtain an elementary proof of the rigidity ofthe hyper-bolic space as the only conformally compact Poincar{\'e}-Einsteinmanifold with the round sphere as its conformal infinity.

A remark on the rigidity of conformally compact Poincar{\'e}-Einstein manifolds