We discuss valuations on convex sets of oriented hyperplanes in R d. For d = 2, we prove an analogue of Hadwiger's characterization theorem for continuous, rigid motion invariant valuations.
It is proved that the shape of the typical cell of a Delaunay tessellation, derived from a stationary Poisson point process in d-dimensional Euclidean space, tends to the shape of a regular simplex, given that the volume of the typical cell tends to infinity. This follows from an estimate for the probability that the typical cell deviates by a given amount… (More)
We study the structure of the space of diametrically complete sets in a finite dimensional normed space. In contrast to the Euclidean case, this space is in general not convex. We show that its starshapedness is equivalent to the completeness of the parallel bodies of complete sets, a property studied in , which is generically not satisfied. The space… (More)
It is proved that the shape of the typical cell of a stationary and isotropic Poisson random hyperplane tessellation is, with high probability, close to the shape of a ball if the kth intrinsic volume (k ≥ 2) of the typical cell is large. The shape of typical cells of large diameter is close to the shape of a segment.