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A generalized lower-bound conjecture for simplicial polytopes

Published online by Cambridge University Press:  26 February 2010

P. McMullen  and
D. W. Walkup
P. McMullen
Affiliation:
University College London. Washington University, St. Louis.
D. W. Walkup
Affiliation:
University College London. Washington University, St. Louis.
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Let P be a simplicial d-polytope, and, for – 1 ≤ j < d, let fj(P) denote the number of j-faces of P (with f_1 (P) = 1). For k = 0, ..., [½d] – 1, we define

and conjecture that

gk(d + 1)(P) ≥ 0,

with equality in the k-th relation if and only if P can be subdivided into a simplicial complex, all of whose simplices of dimension at most d – k – 1 are faces of P. This conjecture is compared with the usual lower-bound conjecture, evidence in support of the conjecture is given, and it is proved that any linear inequality satisfied by the numbers fj(P) is a consequence of the linear inequalities given above.

Type
Research Article
Information
Mathematika , Volume 18 , Issue 2 , December 1971 , pp. 264 - 273
Copyright
Copyright © University College London 1971

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