For which finite subgroups G of SL(r,C), r \geq 4, are there crepant desingularizations of the quotient space C^r/G? A complete answer to this question (also known as "Existence Problem" for such desingularizations) would classify all those groups for which the high-dimensional versions of McKay correspondence are valid. In the paper we consider this question in the case of abelian finite subgroups of SL(r,C) by using techniques from toric and discrete geometry. We give two necessary existence conditions, involving the Hilbert basis elements of the cone supporting the junior simplex, and an Upper Bound Theorem, respectively. Moreover, to the known series of Gorenstein abelian quotient singularities admitting projective, crepant resolutions (which are briefly recapitulated) we add a new series of non-c.i. cyclic quotient singularities having this property.