5 research outputs found
Lower bound theorems for general polytopes
For a -dimensional polytope with vertices, , we
calculate precisely the minimum possible number of -dimensional faces, when
or . This confirms a conjecture of Gr\"unbaum, for these
values of . For , we solve the same problem when or ; the
solution was already known for . 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 -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.Comment: 26 pages, 3 figure