Related papers: Approximation of smooth convex bodies by random po…

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…

Borell's inequality states the existence of a positive absolute constant $C>0$ such that for every $1\leq p\leq q$ $$ \left(\mathbb E|\langle X, e_n\rangle|^p\right)^\frac{1}{p}\leq\left(\mathbb E|\langle X,…

We introduce the vertex index, vein(K), of a given centrally symmetric convex body K, which, in a sense, measures how well K can be inscribed into a convex polytope with small number of vertices. This index is closely connected to the…

We estimate the support of a uniform density, when it is assumed to be a convex polytope or, more generally, a convex body in $\R^d$. In the polytopal case, we construct an estimator achieving a rate which does not depend on the dimension…

We prove a pointwise version of the multi-dimensional central limit theorem for convex bodies. Namely, let X be an isotropic random vector in R^n with a log-concave density. For a typical subspace E in R^n of dimension n^c, consider the…

Let $K\subset\mathbb S^{d-1}$ be a convex spherical body. Denote by $\Delta(K)$ the distance between two random points in $K$ and denote by $\sigma(K)$ the length of a random chord of $K$. We explicitly express the distribution of…

We examine how the measure and the number of vertices of the convex hull of a random sample of $n$ points from an arbitrary probability measure in $\mathbf{R}^d$ relates to the wet part of that measure. This extends classical results for…

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…

We prove Gaussian approximation theorems for specific $k$-dimensional marginals of convex bodies which possess certain symmetries. In particular, we treat bodies which possess a 1-unconditional basis, as well as simplices. Our results…

We consider the following measure of symmetry of a convex n-dimensional body K: $\rho(K)$ is the smallest constant for which there is a point x in K such that for partitions of K by an n-1-dimensional hyperplane passing through x the ratio…

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…

Let $\mathbb{P}_K(n)$ be the probability that $n$ points $z_1,\ldots,z_n$ picked uniformly and independently in $K$, a non-flat compact convex polygon in $\mathbb{R}^2$, are in convex position, that is, form the vertex set of a convex…

We present an affine-invariant random walk for drawing uniform random samples from a convex body $\mathcal{K} \subset \mathbb{R}^n$ that uses maximum volume inscribed ellipsoids, known as John's ellipsoids, for the proposal distribution.…

We experimentally study the fundamental problem of computing the volume of a convex polytope given as an intersection of linear inequalities. We implement and evaluate practical randomized algorithms for accurately approximating the…

It is proved that for a symmetric convex body K in R^n, if for some tau > 0, |K cap (x+tau K)| depends on ||x||_K only, then K is an ellipsoid. As a part of the proof, smoothness properties of convolution bodies ls are studied.

Let $\mu$ be a probability distribution on $\mathbb{R}^d$ which assigns measure zero to every hyperplane and $S$ a set of points sampled independently from $\mu$. What can be said about the expected combinatorial structure of the convex…

The $K$-hull of a compact set $A\subset\mathbb{R}^d$, where $K\subset \mathbb{R}^d$ is a fixed compact convex body, is the intersection of all translates of $K$ that contain $A$. A set is called $K$-strongly convex if it coincides with its…

We prove that for any compact set B in R^d and for any epsilon >0 there is a finite subset X of B of |X|=d^{O(1/epsilon^2)} points such that the maximum absolute value of any linear function ell: R^d --> R on X approximates the maximum…

We approximate boundaries of convex polytopes by smooth hypersurfaces $Y=Y_\varepsilon$ with {\it positive mean curvatures} and, by using basic geometric relations between the scalar curvatures of Riemannin manifolds and the mean curvatures…

Let $K$ be a symmetric convex body in ${\mathbf R}^n$. It is well-known that for every $\theta\in (0,1)$ there exists a subspace $F$ of ${\mathbf R}^n$ with ${\rm dim}F= [(1-\theta )n]$ such that $${\mathcal P}_F(K)\supseteq…