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ON POLYTOPAL UPPER BOUND SPHERES

Part of: Polytopes and polyhedra

Published online by Cambridge University Press:  28 March 2013

Bhaskar Bagchi  and
Basudeb Datta
Bhaskar Bagchi
Affiliation:
Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore 560059,India email bbagchi@isibang.ac.in
Basudeb Datta
Affiliation:
Department of Mathematics, Indian Institute of Science, Bangalore 560012,India email dattab@math.iisc.ernet.in

Abstract

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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 ${ \mathcal{K} }_{k} (2k+ 1)$, i.e., all its vertex links are $k$-stacked spheres. This is surprising since it is far from obvious that the vertex links of polytopal upper bound spheres should have any special combinatorial structure. It has been conjectured that for $d\not = 2k+ 1$, all $(k+ 1)$-neighborly members of the class ${ \mathcal{K} }_{k} (d)$ are tight. The result of this paper shows that the hypothesis $d\not = 2k+ 1$ is essential for every value of $k\geq 1$.

MSC classification

Secondary: 52B11: $n$-dimensional polytopes 52B05: Combinatorial properties (number of faces, shortest paths, etc.) 52B22: Shellability

Information

Type
Research Article
Information
Mathematika , Volume 59 , Issue 2 , July 2013 , pp. 493 - 496
Copyright
Copyright © University College London 2013 

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