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We show that every strictly pseudoconvex domain Ω with smooth boundary in a complex manifold M admits a global defining function, i.e., a smooth plurisubharmonic function φ: U → ℝ defined on an open neighbourhood U ⊂ M of Ω such that Ω = {φ < 0}, dφ ≠ 0 on bΩ and φ is strictly plurisubharmonic near bΩ. We then introduce the notion of the core cΩ of an arbitrary domain Ω ⊂ M} as the set of all points where every smooth and bounded from above plurisubharmonic function on Ω fails to be strictly plurisubharmonic. If Ω is not relatively compact in M, then in general c(Ω) is nonempty, even in the case when M is Stein. It is shown that every strictly pseudoconvex domain Ω ⊂ M with smooth boundary admits a global defining function that is strictly plurisubharmonic precisely in the complement of c(Ω). We then investigate properties of the core. Among other results, we prove 1-pseudoconcavity of the core, we show that in general the core does not possess an analytic structure, and we investigate Liouville type properties of the core.