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On polytopal upper bound spheres

Abstract

Generalizing a result (the case k=1k = 1) due to M. A. Perles, we show that any polytopal upper bound sphere of odd dimension 2k+12k + 1 belongs to the generalized Walkup class Kk(2k+1){\cal K}_k(2k + 1), i.e., all its vertex links are kk-stacked spheres. This is surprising since the kk-stacked spheres minimize the face-vector (among all polytopal spheres with given f0,...,fkβˆ’1f_0,..., f_{k - 1}) while the upper bound spheres maximize the face vector (among spheres with a given f0f_0). It has been conjectured that for dβ‰ 2k+1d\neq 2k + 1, all (k+1)(k + 1)-neighborly members of the class Kk(d){\cal K}_k(d) are tight. The result of this paper shows that, for every kk, the case d=2k+1d = 2k +1 is a true exception to this conjecture.Comment: 4 page

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