arXiv subjects

Michael Joswig

Publications and source records attributed to Michael Joswig.

93 records · Page 6Linked to original sources

arXiv2000

Vertex-Facet Incidences of Unbounded Polyhedra

How much of the combinatorial structure of a pointed polyhedron is contained in its vertex-facet incidences? Not too much, in general, as we demonstrate by examples. However, one can tell from the incidence data whether the polyhedron is bounded. In the case of a polyhedron that is simple and "simplicial," i.e., a d-dimensional polyhedron that has d facets through each vertex and d vertices on each facet, we derive from the structure of the vertex-facet incidence matrix that the polyhedron is necessarily bounded. In particular, this yields a characterization of those polyhedra that have circulants as vertex-facet incidence matrices.

math.CO↗

arXiv1999

Reconstructing a non-simple polytope from its graph

A well-known theorem of Blind and Mani says that every simple polytope is uniquely determined by its graph. Kalai gave a very short and elegant proof of this result using the concept of acyclic orientations. As it turns out, Kalai's proof can be suitably generalized without much effort. We apply our results to a special class of cubical polytopes.

math.CO↗

arXiv1999

Neighborly cubical polytopes

Neighborly cubical polytopes exist: for any $n\ge d\ge 2r+2$, there is a cubical convex d-polytope $C^n_d$ whose $r$-skeleton is combinatorially equivalent to that of the $n$-dimensional cube. This solves a problem of Babson, Billera & Chan. Kalai conjectured that the boundary $\partial C^n_d$ of a neighborly cubical polytope $C^n_d$ maximizes the $f$-vector among all cubical $(d-1)$-spheres with $2^n$ vertices. While we show that this is true for polytopal spheres for $n\le d+1$, we also give a counter-example for $d=4$ and $n=6$. Further, the existence of neighborly cubical polytopes shows that the graph of the $n$-dimensional cube, where $n\ge5$, is ``dimensionally ambiguous'' in the sense of Grünbaum. We also show that the graph of the 5-cube is ``strongly 4-ambiguous''. In the special case $d=4$, neighborly cubical polytopes have $f_3=f_0/4 \log_2 f_0/4$ vertices, so the facet-vertex ratio $f_3/f_0$ is not bounded; this solves a problem of Kalai, Perles and Stanley studied by Jockusch.

math.CO↗