Let $K in R^d$ be a convex body, and assume that $L$ is a randomly rotated and shifted integer lattice. Let $K_L$ be the convex hull of the (random) points $K cap L$. The mean width $W(K_L)$ of $K_L$ is investigated. The asymptotic order of the mean
width difference $W(l K)-W((l K)_L)$ is maximized by the order obtained by polytopes and minimized by the order for smooth convex sets as $l to infty$.
Let $X={x_1,ldots,x_n} subset mathbb R^d$ be an $n$-element point set in general position. For a $k$-element subset ${x_{i_1},ldots,x_{i_k}} subset X$ let the degree ${rm deg}_k(x_{i_1},ldots,x_{i_k})$ be the number of empty simplices ${x_{i_1},ldots
,x_{i_{d+1}}} subset X$ containing no other point of $X$. The $k$-degree of the set $X$, denoted ${rm deg}_k(X)$, is defined as the maximum degree over all $k$-element subset of $X$. We show that if $X$ is a random point set consisting of $n$ independently and uniformly chosen points from a compact set $K$ then ${rm deg}_d(X)=Theta(n)$, improving results previously obtained by Barany, Marckert and Reitzner [Many empty triangles have a common edge, Discrete Comput. Geom., 2013] and Temesvari [Moments of the maximal number of empty simplices of a random point set, Discrete Comput. Geom., 2018] and giving the correct order of magnitude with a significantly simpler proof. Furthermore, we investigate ${rm deg}_k(X)$. In the case $k=1$ we prove that ${rm deg}_1(X)=Theta(n^{d-1})$.
We study the number of facets of the convex hull of n independent standard Gaussian points in d-dimensional Euclidean space. In particular, we are interested in the expected number of facets when the dimension is allowed to grow with the sample size.
We establish an explicit asymptotic formula that is valid whenever d/n tends to zero. We also obtain the asymptotic value when d is close to n.
We provide a random simplicial complex by applying standard constructions to a Poisson point process in Euclidean space. It is gigantic in the sense that - up to homotopy equivalence - it almost surely contains infinitely many copies of every compact
topological manifold, both in isolation and in percolation.
The convex hull $P_{n}$ of a Gaussian sample $X_{1},...,X_{n}$ in $R^{d}$ is a Gaussian polytope. We prove that the expected number of facets $E f_{d-1} (P_n)$ is monotonically increasing in $n$. Furthermore we prove this for random polytopes generat
ed by uniformly distributed points in a $d$-dimensional ball.
The sample range of uniform random points $X_1, dots , X_n$ chosen in a given convex set is the convex hull ${rm conv}[X_1, dots, X_n]$. It is shown that in dimension three the expected volume of the sample range is not monotone with respect to set i
nclusion. This answers a question by Meckes in the negative. The given counterexample is the three-dimensional tetrahedron together with an infinitesimal variation of it. As side result we obtain an explicit formula for all even moments of the volume of a random simplex which is the convex hull of three uniform random points in the tetrahedron and the center of one facet.
Let $X$ be the mosaic generated by a stationary Poisson hyperplane process $hat X$ in ${mathbb R}^d$. Under some mild conditions on the spherical directional distribution of $hat X$ (which are satisfied, for example, if the process is isotropic), we
show that with probability one the set of cells ($d$-polytopes) of $X$ has the following properties. The translates of the cells are dense in the space of convex bodies. Every combinatorial type of simple $d$-polytopes is realized infinitely often by the cells of $X$. A further result concerns the distribution of the typical cell.
Let $Z$ be the typical cell of a stationary Poisson hyperplane tessellation in $mathbb{R}^d$. The distribution of the number of facets $f(Z)$ of the typical cell is investigated. It is shown, that under a well-spread condition on the directional dist
ribution, the quantity $n^{frac{2}{d-1}}sqrt[n]{mathbb{P}(f(Z)=n)}$ is bounded from above and from below. When $f(Z)$ is large, the isoperimetric ratio of $Z$ is bounded away from zero with high probability. These results rely on one hand on the Complementary Theorem which provides a precise decomposition of the distribution of $Z$ and on the other hand on several geometric estimates related to the approximation of polytopes by polytopes with fewer facets. From the asymptotics of the distribution of $f(Z)$, tail estimates for the so-called $Phi$ content of $Z$ are derived as well as results on the conditional distribution of $Z$ when its $Phi$ content is large.
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 an n-dimensional Poisson point process, we study the expected
number of simplices in the Delaunay mosaic as well as the expected number of critical simplices and non-singular 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 get the expected numbers of simplices in the Poisson-Delaunay mosaic in dimensions up to 4.
In a $d$-dimensional convex body $K$ random points $X_0, dots, X_d$ are chosen. Their convex hull is a random simplex. The expected volume of a random simplex is monotone under set inclusion, if $K subset L$ implies that the expected volume of a rand
om simplex in $K$ is smaller than the expected volume of a random simplex in $L$. Continuing work of Rademacher, it is shown that moments of the volume of random simplices are in general not monotone under set inclusion.
