Related papers: The mean width of circumscribed random polytopes

We derive lower estimates for the approximation of the $d$-dimensional Euclidean ball by polytopes with a fixed number of $k$-dimensional faces, $k\in\{0,1,\ldots,d-1\}$. The metrics considered include the intrinsic volume difference and…

Consider a random set of points on the unit sphere in $\mathbb{R}^d$, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the…

Makeev conjectured that every constant-width body is inscribed in the dual difference body of a regular simplex. We prove that homologically, there are an odd number of such circumscribing bodies in dimension 3, and therefore geometrically…

This paper presents bounds for the best approximation, with respect to the Hausdorff metric, of a convex body $K$ by a circumscribed polytope $P$ with a given number of facets. These bounds are of particular interest if $K$ is elongated. To…

We show that the rate of convergence on the approximation of volumes of a convex symmetric polytope P in R^n by its dual L_{p$-centroid bodies is independent of the geometry of P. In particular we show that if P has volume 1,…

If K' and K are convex bodies of the plane such that K' is a subset of K then the perimeter of K' is not greater than the perimeter of K. We obtain the following generalization of this fact. Let K be a convex compact body of the plane with…

We consider the convex hull of a finite sample of i.i.d. points uniformly distributed in a convex body in $\R^d$, $d\geq 2$. We prove an exponential deviation inequality, which leads to rate optimal upper bounds on all the moments of the…

Facets of the convex hull of $n$ independent random vectors chosen uniformly at random from the unit sphere in $\mathbb{R}^d$ are studied. A particular focus is given on the height of the facets as well as the expected number of facets as…

Let K be a convex set in R d and let K $\lambda$ be the convex hull of a homogeneous Poisson point process P $\lambda$ of intensity $\lambda$ on K. When K is a simple polytope, we establish scaling limits as $\lambda$ $\rightarrow$ $\infty$…

Let $K$ be a convex body in $\Bbb R^{d}$ and $K_{t}$ its floating bodies. There is a polytope with at most $n$ vertices that satisfies $$ K_{t} \subset P_{n} \subset K $$ where $$ n \leq e^{16d} \frac{vol_{d}(K \setminus K_{t})}{t\…

The deviation of a general convex body with twice differentiable boundary and an arbitrarily positioned polytope with a given number of vertices is studied. The paper considers the case where the deviation is measured in terms of the…

The beta polytope $P_{n,d}^\beta$ is the convex hull of $n$ i.i.d. random points distributed in the unit ball of $\mathbb{R}^d$ according to a density proportional to $(1-\lVert{x}\rVert^2)^{\beta}$ if $\beta>-1$ (in particular, $\beta=0$…

For a convex body $K \subset {\mathbb R}^n$, let $K^z = \{y\in{\mathbb R}^n : \langle y-z, x-z\rangle\le 1, \mbox{\ for all\ } x\in K\}$ be the polar body of $K$ with respect to the center of polarity $z \in {\mathbb R}^n$. The goal of this…

For a finite point set $P \subset \mathbb{R}^d$, denote by $\text{diam}(P)$ the ratio of the largest to the smallest distances between pairs of points in $P$. Let $c_{d, \alpha}(n)$ be the largest integer $c$ such that any $n$-point set $P…

Recall that a convex body $K$ is in John's position if the unit Euclidean ball is the maximal volume ellipsoid contained in $K$. Approximating convex body in John's position by polytopes we obtain the following results. 1. Let $n>R_n\ge 1$…

An equidistant polytope is a special equidistant set in the space $\mathbb{R}^n$ all of whose boundary points have equal distances from two finite systems of points. Since one of the finite systems of the given points is required to be in…

Let $p_n$ denote the number of self-avoiding polygons of length $n$ on a regular three-dimensional lattice, and let $p_n(K)$ be the number which have knot type $K$. The probability that a random polygon of length $n$ has knot type $K$ is…

We study the geometry of centrally-symmetric random polytopes, generated by $N$ independent copies of a random vector $X$ taking values in $\mathbb{R}^n$. We show that under minimal assumptions on $X$, for $N \gtrsim n$ and with high…

Fix a space dimension $d\ge 2$, parameters $\alpha > -1$ and $\beta \ge 1$, and let $\gamma_{d,\alpha, \beta}$ be the probability measure of an isotropic random vector in $\mathbb{R}^d$ with density proportional to \begin{align*}…

This paper considers the question of how to succinctly approximate a multidimensional convex body by a polytope. Given a convex body $K$ of unit diameter in Euclidean $d$-dimensional space (where $d$ is a constant) and an error parameter…