2,314,775 research outputs found
A Geometric Lower Bound Theorem
We resolve a conjecture of Kalai relating approximation theory of convex
bodies by simplicial polytopes to the face numbers and primitive Betti numbers
of these polytopes and their toric varieties. The proof uses higher notions of
chordality. Further, for C^2-convex bodies, asymptotically tight lower bounds
on the g-numbers of the approximating polytopes are given, in terms of their
Hausdorff distance from the convex body.Comment: 26 pages, 6 figures, to appear in Geometric and Functional Analysi
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
- …