Abstract.
Consider the question: Given integers 0 \le k<d<n , does there exist a simple d -polytope with n faces of dimension k ? We show that there exist numbers G(d,k) and N(d,k) such that for n> N(d,k) the answer is yes if and only if {G(d,k)} divides n . Furthermore, a formula for G(d,k) is given, showing that, e.g., G(d,k)=1 if \( k \ge \left\lfloor (d+1)/{2}\right\rfloor \) or if both d and k are even, and also in some other cases (meaning that all numbers beyond N(d,k) occur as the number of k -faces of some simple d -polytope).
This question has previously been studied only for the case of vertices (k=0 ), where Lee [Le] proved the existence of N(d,0) (with G(d,0)=1 or 2 depending on whether d is even or odd), and Prabhu [P1] showed that \( N(d,0) \le cd\sqrt {d} \) . We show here that asymptotically the true value of Prabhu's constant is \( c=\sqrt2 \) if d is even, and c=1 if d is odd.
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Received December 16, 1996, and in revised form March 4, 1997.
An erratum to this article is available at 10.1007/s00454-007-0382-3.
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Björner, A., Linusson, S. The Number of k -Faces of a Simple d -Polytope . Discrete Comput Geom 21, 1–16 (1999). https://doi.org/10.1007/PL00009403
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DOI: https://doi.org/10.1007/PL00009403