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The concept of typical and weighted typical spherical faces for tessellations of the $d$-dimensional unit sphere, generated by $n$ independent random great hyperspheres distributed according to a non-degenerate directional distribution, is introduced and studied. Probabilistic interpretations for such spherical faces are given and their directional distributions are determined. Explicit formulas for the expected $f$-vector, the expected spherical Quermass integrals and the expected spherical intrinsic volumes are found in the isotropic case. Their limiting behaviour as $ntoinfty$ is discussed and compared to the corresponding notions and results in the Euclidean case. The expected statistical dimension and a problem related to intersection probabilities of spherical random polytopes is investigated.
Curvature is a fundamental geometric characteristic of smooth spaces. In recent years different notions of curvature have been developed for combinatorial discrete objects such as graphs. However, the connections between such discrete notions of curv
Let $xi_1,xi_2,ldots$ be a sequence of independent copies of a random vector in $mathbb R^d$ having an absolutely continuous distribution. Consider a random walk $S_i:=xi_1+cdots+xi_i$, and let $C_{n,d}:=text{conv}(0,S_1,S_2,ldots,S_n)$ be the convex
In this paper, we construct a new family of random series defined on $R^D$, indexed by one scaling parameter and two Hurst-like exponents. The model is close to Takagi-Knopp functions, save for the fact that the underlying partitions of $R^D$ are not
We study a geometric property related to spherical hyperplane tessellations in $mathbb{R}^{d}$. We first consider a fixed $x$ on the Euclidean sphere and tessellations with $M gg d$ hyperplanes passing through the origin having normal vectors distrib
Let $V$ be a highest weight module over a Kac-Moody algebra $mathfrak{g}$, and let conv $V$ denote the convex hull of its weights. We determine the combinatorial isomorphism type of conv $V$, i.e. we completely classify the faces and their inclusions