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Mapping every simplex in the Delaunay mosaic of a discrete point set to the radius of the smallest empty circumsphere gives a generalized discrete Morse function. Choosing the points from a Poisson point process in ℝn, we study the expected number of simplices in the Delaunay mosaic as well as the expected number of critical simplices and nonsingular intervals in the corresponding generalized discrete gradient. Observing connections with other probabilistic models, we obtain precise expressions for the expected numbers in low dimensions. In particular, we obtain the expected numbers of simplices in the Poisson–Delaunay mosaic in dimensions n ≤ 4.
-dimensional convex body
are chosen. Their convex hull is a random simplex. The expected volume of a random simplex is monotone under set inclusion if
implies that the expected volume of a random simplex in
is smaller than the expected volume of a random simplex in
. Continuing work of Rademacher [On the monotonicity of the expected volume of a random simplex. Mathematika58 (2012), 77–91], it is shown that moments of the volume of random simplices are, in general, not monotone under set inclusion.
The convex hull of n independent random points in ℝd, chosen according to the normal distribution, is called a Gaussian polytope. Estimates for the variance of the number of i-faces and for the variance of the ith intrinsic volume of a Gaussian polytope in ℝd, d∈ℕ, are established by means of the Efron-Stein jackknife inequality and a new formula of Blaschke-Petkantschin type. These estimates imply laws of large numbers for the number of i-faces and for the ith intrinsic volume of a Gaussian polytope as n→∞.
A random polytope is the convex hull of n random points in the interior of a convex body K. The expectation of the ith intrinsic volume of a random polytope as n → ∞ is investigated. It is proved that, for convex bodies of differentiability class Kk+1, precise asymptotic expansions for these expectations exist. The proof makes essential use of a refinement of Crofton's boundary theorem.
In a convex domain K in ℝd, a transmitter and a receiver are placed at random according to the uniform distribution. The statistics of the power received by the receiver is an important quantity for the design of wireless communication systems. Bounds for the moments of the received power are given, which depend only on the volume and the surface area of the convex domain.
It is proved that the shape of the typical cell of a stationary Poisson-Voronoi tessellation in Euclidean space, under the condition that the volume of the typical cell is large, must be close to spherical, with high probability. The same holds if the volume is replaced by the surface area or other suitable functionals. Similar results are established for the zero cell of a stationary and isotropic Poisson hyperplane tessellation.
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