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Random Polytopes

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Stochastic Geometry, Spatial Statistics and Random Fields

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2068))

Abstract

Random polytopes arise naturally as convex hulls of random points selected according to a given distribution. In a dual way, they can be derived as intersections of random halfspaces. Still another route to random polytopes is via the consideration of special cells and faces associated with random mosaics. The study of random polytopes is based on the fruitful interplay between geometric and probabilistic methods. This survey describes some of the geometric concepts and arguments that have been developed and applied in the context of random polytopes. Among these are duality arguments, geometric inequalities and stability results for various geometric functionals, associated bodies and zonoids as well as methods of integral geometry. Particular emphasis is given to results on the shape of large cells in random tessellations, as suggested in Kendall’s problem.

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  1. Karlsruhe Institute of Technology, Karlsruhe, Germany

    Daniel Hug

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    Evgeny Spodarev

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Hug, D. (2013). Random Polytopes. In: Spodarev, E. (eds) Stochastic Geometry, Spatial Statistics and Random Fields. Lecture Notes in Mathematics, vol 2068. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33305-7_7

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