Inner diagonals of convex polytopes

by 

David Bremner, 
Department of Mathematics, University of Washington

Abstract:

An {\it inner diagonal} of a polytope $P$ is a segment that joins two
vertices of $P$ and that lies, except for its ends, in $P$'s relative
interior. A tantalizing conjecture due to von Stengel claims that
among simple $d$-polytopes with $2d$ facets, the maximum number of
inner diagonals is achieved by a $d$-cube.  In this talk I will
present a characterization of the maximum and minimum number of inner
diagonals achievable in  3 dimensions for fixed numbers of vertices
or facets.  I will also present partial results in higher dimensions,
including an interesting relationship to Kalai's new proof (based on
rigidity of graphs) of the Lower Bound Theorem.

This is joint work with Victor Klee.