Generalizing a result (the case k=1) due to M. A. Perles, we show that
any polytopal upper bound sphere of odd dimension 2k+1 belongs to the
generalized Walkup class Kkβ(2k+1), i.e., all its vertex links are
k-stacked spheres. This is surprising since the k-stacked spheres minimize
the face-vector (among all polytopal spheres with given f0β,...,fkβ1β)
while the upper bound spheres maximize the face vector (among spheres with a
given f0β).
It has been conjectured that for dξ =2k+1, all (k+1)-neighborly
members of the class Kkβ(d) are tight. The result of this paper shows
that, for every k, the case d=2k+1 is a true exception to this
conjecture.Comment: 4 page