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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)
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 [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)