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Two new approaches are presented to establish the existence of polytopal solutions to the discrete-data L p Minkowski problem for all p > 1. As observed by Schneider [21], the Brunn-Minkowski theory springs from joining the notion of ordinary volume in Euclidean d-space, R d , with that of Minkowski combinations of convex bodies. One of the cornerstones of… (More)

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)

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.

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.

Let K be a convex body in R d , let j ∈ {1,. .. , d − 1}, and let K(n) be the convex hull of n points chosen randomly, independently and uniformly from K. If ∂K is C 2 + , then an asymptotic formula is known due to M. Reitzner (and due to I. Bárány if ∂K is C 3 +) for the difference of the jth intrinsic volume of K and the expectation of the jth intrinsic… (More)

A random spherical polytope P n in a spherically convex set K ⊂ S d as considered here is the spherical convex hull of n independent, uniformly distributed random points in K. The behaviour of P n for a spherically convex set K contained in an open halfsphere is quite similar to that of a similarly generated random convex polytope in a Euclidean space, but… (More)

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