McMullen's conditions and some lower bounds for general convex polytopes
In the combinatorial theory of convex polytopes an important question is the connection between the numbers of the \(i\)-faces of a \(d\)-dimensional polytope \((i = 0,1,\dots, d-1)\). In particular 0-faces, 1-faces and \((d - 1)\)-faces are respectively called vertices, edges and facets of a \(d\)- dimensional polytope; \(f_ i(P)\) denotes the number of the \(i\)-faces of a polytope \(P\). Especially upper and lower bounds of these numbers \(f_ i(P)\) for \(P\) with a given number of \(k\)-faces are considered. Furthermore, one wants to know which classes of polytopes attain such bounds. \textit{P. McMullen's} upper bound theorem [Mathematika, London 17, 179-184 (1970; Zbl 0217.467)] gives upper bounds for the numbers \(f_ i(P)\) \((i = 0,1, \dots, d-1)\) for the set of all \(d\)-dimensional polytopes \(P\) with \(m\) vertices. These bounds are attained for special simplicial polytopes. \textit{D. Barnette's} lower bound theorem [Pac. J. Math. 46, 349-354 (1973; Zbl 0264.52006)] concerns sharp lower bounds of the numbers \(f_ i (P)\) (\(i = 0,1, \dots, d-1\)) only for \(d\)-dimensional simplicial polytopes \(P\) with \(m\) vertices. In this note the authors present lower bounds \(\ell^ d_ i (m)\) of \(f_ 0 (P)\) for a general \(d\)-dimensional polytope \(P\) with \(M\) \(i\)- faces \((i = 0,1, \dots, \lfloor d/2\rfloor - 1)\). These bounds are attained for special simplicial polytopes and are a sort of inverse functions of McMullen's bounds. The authors mention the asymptotical representation \(\ell^ d_ i (m) = O(\root{i+1} \of {m}\)). To show that \(\ell^ d_ i(m)\) is a lower bound the authors use Barnette's results. For tightness of these bounds McMullen's conditions are utilized.
scientific article; zbMATH DE number 679539
scientific article