14 research outputs found
One-Point Suspensions and Wreath Products of Polytopes and Spheres
It is known that the suspension of a simplicial complex can be realized with
only one additional point. Suitable iterations of this construction generate
highly symmetric simplicial complexes with various interesting combinatorial
and topological properties. In particular, infinitely many non-PL spheres as
well as contractible simplicial complexes with a vertex-transitive group of
automorphisms can be obtained in this way.Comment: 17 pages, 8 figure
An update on the Hirsch conjecture
The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to
George Dantzig. It states that the graph of a d-dimensional polytope with n
facets cannot have diameter greater than n - d.
Despite being one of the most fundamental, basic and old problems in polytope
theory, what we know is quite scarce. Most notably, no polynomial upper bound
is known for the diameters that are conjectured to be linear. In contrast, very
few polytopes are known where the bound is attained. This paper collects
known results and remarks both on the positive and on the negative side of the
conjecture. Some proofs are included, but only those that we hope are
accessible to a general mathematical audience without introducing too many
technicalities.Comment: 28 pages, 6 figures. Many proofs have been taken out from version 2
and put into the appendix arXiv:0912.423