We analyze the gravity-induced effects associated with a massless scalar field in a higher-dimensional spacetime being the tensor product of (d - n)-dimensional Minkowski space and n-dimensional spherically/cylindrically-symmetric space with a solid/planar angle deficit. In particular, we revisit the computation of the vacuum polarization effects for a non-minimally coupled massless scalar field in the spacetime of a straight cosmic string. We compute the renormalized vacuum expectation value of the field square <Φ~2(x)>_(ren) and the renormalized vacuum averaged of the scalar-field's energy-momentum tensor _(ren). The explicit dependence of the results upon the dimensionalities of both the bulk and conical submanifold and upon the coupling constant to the curvature ξ is discussed.
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