The upper bound inequality h i (P)&h i&1 (P) ( n&d+i&2 i ) (0 i dΓ2) is proved for the toric h-vector of a rational convex d-dimensional polytope with n vertices. This gives nonlinear inequalities on flag vectors of rational polytopes.
1998 Academic Press
A major result in polytope theory is the characterization of face vectors of simplicial polytopes, conjectured by McMullen [10] and proved by Stanley (necessity [11]) and Billera and Lee (sufficiency [3]). The ``McMullen conditions'' in the theorem are stated most easily in terms of the h-vector of the polytope. For P a simplicial d-polytope, write f i (P) for the number of i-faces of P, and define the h-vector, h(P)=(h 0 , h 1 , ..., h d ) by h j = j i=0 (&1) j&i ( d&i d&j ) f i&1 . Write g i =h i &h i&1 for 1 i dΓ2; g 0 =h 0 . For the definition of the nonlinear operator (i ), see, for example, [11].
Theorem (Stanley, Billera, and Lee). An integer vector (h 0 , h 1 , ..., h d ) is the h-vector of a simplicial polytope if and only if (i) h i =h d&i for all i (ii) g 0 =1, and g i 0 for 1 i dΓ2 (iii) g i+1 ( g i ) ( i) for 0 i dΓ2&1.
We present the Upper Bound Theorem as a corollary. McMullen [9] proved parts (ii) and (iii) below ten years before Stanley proved the necessity of the McMullen conditions and hence part (i) of the corollary.