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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)
For a given convex body K in R d , a random polytope K (n) is defined (essentially) as the intersection of n independent closed halfspaces containing K and having an isotropic and (in a specified sense) uniform distribution. We prove upper and lower bounds, of optimal orders, for the difference of the mean widths of K (n) and K, as n tends to infinity. For(More)
BACKGROUND AND PURPOSE Genetic risk factors are thought to play a role in the etiology of spontaneous cervical artery dissections (CAD). However, familial CAD is extremely rare. In this study we analyzed patients with familial CAD and asked the question whether familial CAD has particular features. METHODS Seven families with 15 CAD patients were(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)