%0 Journal Article
%T On polytopal upper bound spheres
%A Bhaskar Bagchi
%A Basudeb Datta
%J Mathematics
%D 2012
%I arXiv
%R 10.1112/S0025579313000016
%X 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.
%U http://arxiv.org/abs/1207.5098v1