  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 ${\cal K}_k(2k + 1)$, 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 $f_0,..., f_{k - 1}$)
while the upper bound spheres maximize the face vector (among spheres with a
given $f_0$).
  It has been conjectured that for $d\neq 2k + 1$, all $(k + 1)$-neighborly
members of the class ${\cal K}_k(d)$ are tight. The result of this paper shows
that, for every $k$, the case $d = 2k +1$ is a true exception to this
conjecture.
