Lower bound theorems for general polytopes

TL;DR

This paper determines the minimum number of faces for certain polytopes with given vertices, confirming a conjecture and revealing gaps in face counts, advancing the understanding of polytope face enumeration.

Contribution

It provides exact lower bounds for face counts of polytopes with specific vertex counts, confirming Grünbaum's conjecture for certain cases, and characterizes the polytopes achieving these bounds.

Findings

Exact minimum face counts for polytopes with v vertices where d+1 ≤ v ≤ 2d.

Confirmation of Grünbaum's conjecture for m=1 and m≥0.62d.

Identification of gaps in possible face counts, e.g., no polytope with 80 edges in dimension 10.

Abstract

For a $d$-dimensional polytope with $v$ vertices, $d+1\le v\le2d$, we calculate precisely the minimum possible number of $m$-dimensional faces, when $m=1$ or $m\ge0.62d$. This confirms a conjecture of Gr\"unbaum, 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.