We establish a close relationship between isoperimetric inequalities for convex bodies and asymptotic shapes of large random polytopes, which arise as cells in certain random mosaics in d-dimensional Euclidean space. These mosaics are generated by Poisson hyperplane processes satisfying a few natural assumptions (not necessarily sta-tionarity or isotropy).… (More)
The principal kinematic formula and the closely related Crofton formula are central themes of integral geometry in the sense of Blaschke and Santaló. There have been various generalizations , variants, and analogues of these formulae, in part motivated by applications. We give a survey of recent investigations in the spirit of the kinematic and Crofton… (More)
We consider geometric functionals of the convex hull of normally distributed random points in Eu-clidean space R d. In particular, we determine the asymptotic behaviour of the expected value of such functionals and of related geometric probabilities, as the number of points increases.
This paper originates from the investigation of support measures of convex bodies (sets of positive reach), which form a central subject in convex geometry and also represent an important tool in related fields. We show that these measures are absolutely continuous with respect to Hausdorff measures of appropriate dimensions, and we determine the… (More)
For a given convex (semi-convex) function u, defined on a nonempty open convex set Ω ⊂ R n , we establish a local Steiner type formula , the coefficients of which are nonnegative (signed) Borel measures. We also determine explicit integral representations for these coefficient measures, which are similar to the integral representations for the curvature… (More)
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 , 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.
We prove a complete set of integral geometric formulas of Crofton type (involving in-tegrations over affine Grassmannians) for the Minkowski tensors of convex bodies. Min-kowski tensors are the natural tensor valued valuations generalizing the intrinsic volumes (or Minkowski functionals) of convex bodies. By Hadwiger's general integral geometric theorem,… (More)