Lower bound theorems for general polytopes

Abstract

For a $d$-dimensional polytope with $v$ vertices, $d+1\le v\le 2d$, we calculate precisely the minimum possible number of $m$-dimensional faces, when $m=1$ or $m\ge 0.62d$. This confirms a conjecture of Grünbaum, for these values of $m$. For $v=2d+1$, we solve the same problem when $m=1$ or $d-2$; the solution was already known for $m=d-1$. In all these cases, we give a characterisation of the minimising polytopes. We also show that there are many gaps in the possible number of $m$-faces: for example, there is no polytope with 80 edges in dimension 10, and a polytope with 407 edges can have dimension at most 23.

26 pages, 3 figures

References in corpus (1)

• More indecomposable polyhedra

Cited by in corpus (4)

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