406 research outputs found
Elementary moves on triangulations
It is proved that a triangulation of a polyhedron can always be transformed
into any other triangulation of the polyhedron by using only elementary moves.
One consequence is that an additive function (valuation) defined only on
simplices may always be extended to an additive function on all polyhedra.Comment: 10 pages; some corrections, extended proofs of Lemma 4 and Corollary
U-statistics in stochastic geometry
This survey will appear as a chapter of the forthcoming book [19]. A
U-statistic of order with kernel f:\X^k \to \R^d over a Poisson process
is defined in \cite{ReiSch11} as under appropriate integrability
assumptions on . U-statistics play an important role in stochastic geometry
since many interesting functionals can be written as U-statistics, like
intrinsic volumes of intersection processes, characteristics of random
geometric graphs, volumes of random simplices, and many others, see for
instance \cite{ LacPec13, LPST,ReiSch11}. It turns out that the Wiener-Ito
chaos expansion of a U-statistic is finite and thus Malliavin calculus is a
particularly suitable method. Variance estimates, the approximation of the
covariance structure and limit theorems which have been out of reach for many
years can be derived. In this chapter we state the fundamental properties of
U-statistics and investigate moment formulae. The main object of the chapter is
to introduce the available limit theorems.Comment: 22p
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