Large convex sets in oriented matroids

On Delaunay Oriented Matroids for Convex Distance Functions

For any nite point set S in Ed, an oriented matroid DOM(S) can be de ned in terms of how S is partitioned by Euclidean hyperspheres. This oriented matroid is related to the Delaunay triangulation of S and is realizable, because of the lifting property of Delaunay triangulations. We prove that the same construction of a Delaunay oriented matroid can be performed with respect to any smooth, stric...

Fachbereich 3 Mathematik Oriented Matroids Oriented Matroids

Valuations on Convex Sets of Oriented Hyperplanes

We discuss valuations on convex sets of oriented hyperplanes in R. For d = 2, we prove an analogue of Hadwiger’s characterization theorem for continuous, rigid motion invariant valuations.

Large convex holes in random point sets

A convex hole (or empty convex polygon) of a point set P in the plane is a convex polygon with vertices in P , containing no points of P in its interior. Let R be a bounded convex region in the plane. We show that the expected number of vertices of the largest convex hole of a set of n random points chosen independently and uniformly over R is Θ(logn/(log log n)), regardless of the shape of R.

Matroids on convex geometries (cg-matroids)

We consider matroidal structures on convex geometries, which we call cg-matroids. The concept of a cg-matroid is closely related to but different from that of a supermatroid introduced by Dunstan, Ingleton, and Welsh in 1972. Distributive supermatroids or poset matroids are supermatroids defined on distributive lattices or sets of order ideals of posets. The class of cg-matroids includes distri...