Related papers: Cotype of random polytopes

We define certain natural finite sums of $n$'th roots of unity, called $G_P(n)$, that are associated to each convex integer polytope $P$, and which generalize the classical $1$-dimensional Gauss sum $G(n)$ defined over $\mathbb Z/ {n…

Random simplices and more general random convex bodies of dimension $p$ in $\mathbb{R}^n$ with $p\leq n$ are considered, which are generated by random vectors having an elliptical distribution. In the high-dimensional regime, that is, if…

We establish central limit theorems for natural volumes of random inscribed polytopes in projective Riemannian or Finsler geometries. In addition, normal approximation of dual volumes and the mean width of random polyhedral sets are…

Let C = C(l_1, ..., l_n) be the n-dimensional orthogonal cross-polytope whose axes are of length l_1,..., l_n. Subject to the condition \sum l_i^2 = 1, the mean width of C is minimised when l_i = 1/sqrt{n} for every i, and it is maximised…

We prove the following isoperimetric-type inequality: for every convex body $K$ in $\mathbb R^n$ and some $\sigma\subset[n]:=\{1,\dots,n\}$ there exists a suitable Hanner polytope $B_K$ with the same volume as $K$ and such that the volume…

Let $X_1,\ldots,X_n$ be a standard normal sample in $\mathbb R^d$. We compute exactly the expected volume of the Gaussian polytope $\mathrm{conv}[X_1,\ldots,X_n]$, the symmetric Gaussian polytope $\mathrm{conv}[\pm X_1,\ldots,\pm X_n]$, and…

The type-PQ adjacency polytope associated to a simple graph is a $0/1$-polytope containing valuable information about an underlying power network. Chen and the first author have recently demonstrated that, when the underlying graph $G$ is…

While there is extensive literature on approximation, deterministic as well as random, of general convex bodies $K$ in the symmetric difference metric, or other metrics arising from intrinsic volumes, very little is known for corresponding…

Let X_{d,n} be an n-element subset of {0,1}^d chosen uniformly at random, and denote by P_{d,n} := conv X_{d,n} its convex hull. Let D_{d,n} be the density of the graph of P_{d,n} (i.e., the number of one-dimensional faces of P_{d,n}…

In this paper, we deal with the question; under what conditions the points $P_i(xi,yi)$ $(i = 1,\cdots, n)$ form a convex polygon provided $x_1 < \cdots < x_n$ holds. One of the main findings of the paper can be stated as follows: "Let…

We study cosmological polytopes induced by Erd\H{o}s--R\'enyi random graphs in a high-dimensional regime. These graph-based lattice polytopes form a natural model of random lattice polytopes in which geometric features are determined by the…

It was shown in \cite{GL} that the maximal surface area of a convex set in $\mathbb{R}^n$ with respect to a rotation invariant log-concave probability measure $\gamma$ is of order $\frac{\sqrt{n}}{\sqrt[4]{Var|X|} \sqrt{\mathbb{E}|X|}}$,…

Two models of random cones in high dimensions are considered, together with their duals. The Donoho-Tanner random cone $D_{n,d}$ can be defined as the positive hull of $n$ independent $d$-dimensional Gaussian random vectors. The Cover-Efron…

Let $P_1,\dots, P_n$ and $Q_1,\dots, Q_n$ be convex polytopes in $\mathbb{R}^n$ such that $P_i\subset Q_i$. It is well-known that the mixed volume has the monotonicity property: $V(P_1,\dots,P_n)\leq V(Q_1,\dots,Q_n)$. We give two criteria…

Let K be a closed bounded convex subset of $\Bbb R^n$; then by a result of the first author, which extends a classical theorem of Whitney there is a constant $w_m(K)$ so that for every continuous function f on K there is a polynomial $\phi$…

The cross covariogram g_{K,L} of two convex sets K, L in R^n is the function which associates to each x in R^n the volume of the intersection of K with L+x. The problem of determining the sets from their covariogram is relevant in…

Let $(k_n)_{n \in \mathbb{N}}$ be a sequence of positive integers growing to infinity at a sublinear rate, $k_n \rightarrow \infty$ and $k_n/n \rightarrow 0$ as $n \rightarrow \infty$. Given a sequence of $n$-dimensional random vectors…

Let $R$ be a polynomial ring in $N$ variables over an arbitrary field $K$ and let $I$ be an ideal of $R$ generated by $n$ polynomials of degree at most 2. We show that there is a bound on the projective dimension of $R/I$ that depends only…

Let $E$ be a Banach space and, for any positive integer $n$, let ${\cal P}(^nE)$ denote the Banach space of continuous $n$-homogeneous polynomials on $E$. Davie and Gamelin showed that the natural extension mapping from ${\cal P}(^nE)$ to…

A question related to some conjectures of Lutwak about the affine quermassintegrals of a convex body $K$ in ${\mathbb R}^n$ asks whether for every convex body $K$ in ${\mathbb R}^n$ and all $1\leqslant k\leqslant n$ $$\Phi_{[k]}(K):={\rm…