On the coordinatization of oriented matroids

Several important and hard realizability problems of combinatorial geometry can be reduced to the realizability problem of oriented matroids. In this paper we describe a method to find a coordinatization for a large class of realizable cases. This algorithm has been used successfully to decide several geometric realizability problems. It is shown that all realizations found by our algorithm fulfill the isotopy property.     

Integral representations of quermassintegrals and Bonnesen-style inequalities

Archiv der Mathematik -     

A geometric realization without self-intersections does exist for Dyck's regular map

Even in our decade there is still an extensive search for analogues of the Platonic solids. In a recent paper Schulte and Wills [13] discussed properties of Dyck's regular map of genus 3 and gave polyhedral realizations for it allowing self-intersections. This paper disproves their conjecture in showing that there is a geometric polyhedral realization (without self-intersections) of Dyck's regular map {3, 8}₆ already in Euclidean 3-space. We describe the shape of this new regular polyhedron.     

On the cube problem of Las Vergnas

We prove a conjecture of Las Vergnas in dimensions d7: The matroid of the d-dimensional cube C ^d has a unique reorientation class. This extends a result of Las Vergnas, Roudneff and Salaün in dimension 4. Moreover, we determine the automorphism group G ^d of the matroid of the d-cube C ^d for arbitrary dimension d, and we discuss its relation to the Coxeter group of C ^d . We introduce matroid facets of the matroid of the d-cube in order to evaluate the order of G ^d . These matroid facets turn out to be arbitrary pairs of parallel subfacets of the cube. We show that the Euclidean automorphism group W ^d is a proper subgroup of the group G ^d of all matroid symmetries of the d-cube by describing genuine matroid symmetries for each Euclidean facet. A main theorem asserts that any one of these matroid symmetries together with the Euclidean Coxeter symmetries generate the full automorphism group G ^d . For the proof of Las Vergnas' conjecture we use essentially these symmetry results together with the fact that the reorientation class of an oriented matroid is determined by the labeled lower rank contractions of the oriented matroid. We also describe the Folkman-Lawrence representation of the vertex figure of the d-cube and a contraction of it. Finally, we apply our method of proof to show a result of Las Vergnas, Roudneff, and Salaün that the matroid of the 24-cell has a unique reorientation class, too.     

A polyhedron of genus 4 with minimal number of vertices and maximal symmetry

We construct a symmetric polyhedron of genus 4 in R ³ with 11 vertices. This shows that for given genus g the minimal numbers of vertices of combinatorial manifolds and of polyhedra coincide in the first previously unknown case g=4 also. We show that our polyhedron has the maximal symmetry for the given genus and minimal number of vertices.     

Dichte Klassen konvexer Polytope

Ohne Zusammenfassung     

On the Sylvester–Gallai and the orchard problem for pseudoline arrangements

We study a non-trivial extreme case of the orchard problem for 12 pseudolines and we provide a complete classification of pseudoline arrangements having 19 triple points and 9 double points. We have also classified those that can be realized with straight lines. They include new examples different from the known example of Böröczky. Since Melchior’s inequality also holds for arrangements of pseudolines, we are able to deduce that some combinatorial point-line configurations cannot be realized using pseudolines. In particular, this gives a negative answer to one of Grünbaum’s problems. We formulate some open problems which involve our new examples of line arrangements.     

Symmetric matroid polytopes and their generation

Matroid polytopes form an intermediate structure useful in searching for realizable convex spheres. In this article we present a class of self-polar 3-spheres that motivated research in the inductive generation of matroid polytopes, along with two new methods of generation.     

Equifacetted 3-spheres as topes of nonpolytopal matroid polytopes

We investigate simplicial 3-manifolds, in particular 3-spheres, with few vertices such that the links of all vertices are combinatorially equivalent (equilinked 3-spheres), and, simple 3-manifolds, in particular 3-spheres, with few facets such that all facets are combinatorially equivalent (equifacetted 3-spheres).     

On representing contexts in line arrangements

Acontext is defined to be a triple (G, M, J) of setsG, M and an incidence relationJ G×M.A finite set ofn oriented lines in general position in the euclidean plane induces a cell decomposition of the plane. For a givenk-element subset of cells of dimension 2, we define an incidence relationJ × as follows:t _i andl _j are incident if and only ift _i lies on the positive side with respect tol _j .We call a context (G, M, J)represented in a line arrangement if and only if there are relation preserving bijections betweenG and ,M and , respectively. We study conditions for a context to be representable in a line arrangement.Especially, we provide a non-trivial infinite class of contexts which can not be represented in a line arrangement.