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We investigate the conjectured sufficiency of a condition for h-vectors (1, h1, h2, . . . , hd, 0) of regular d-dimensional triangulations. (The condition is already shown to be necessary in [2]). We first prove that the condition is sufficient when h1 ≥ h2 ≥ · · · ≥ hd. We then derive some new shellings of squeezed spheres and use them to prove that the condition is sufficient when d = 3. Finally, in the case d = 4, we construct shellable 4-balls with the desired h-vectors, showing them to be realizable as regular triangulations when h4 = 0 or h4 = h1.

On the numbers of faces of low-dimensional regular triangulations and shellable balls