The Number of k -Faces of a Simple d -Polytope
Tóm tắt
Consider the question: Given integers 0 \le
k 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.