We investigate the conjectured sufficiency of a condition for h-vectors (1, h 1 , h 2 , . . . , h d , 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 h 1 ≥ h 2 ≥ · · · ≥ h d . 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

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... em to be realizable as regular triangulations when h 4 = 0 or h 4 = h 1 .