Related papers: Concentration of random polytopes around the expec…

We show that the Newton polytope of a polynomial has a strong impact on the distribution of its mass and zeros. The basic theme is that Newton polytopes determine allowed and forbidden regions for these distributions. We equip the space of…

We prove a strong law of large numbers for random sets with bounded and closed values contained in an arbitrary (not necessarily separable) Banach space. We make use of a notion of convergence of sets introduced by Fisher, which is stronger…

In (the surface of) a convex polytope P^n in R^n+1, for small prescribed volume, geodesic balls about some vertex minimize perimeter. This revision corrects a mistake in the mass bound argument in the proof of Theorem 3.8.

A new version of a strong law of large numbers for a ``good'' pairwise independent sequence of random variables (r.v.'s) with a small part of ``bad'' dependent r.v.'s is proposed. The main goal is to relax the assumption on the existence of…

An identity due to Efron dating from 1965 relates the expected volume of the convex hull of $n$ random points to the expected number of vertices of the convex hull of $n+1$ random points. Forty years later this identity was extended from…

We investigate weighted floating bodies of polytopes. We show that the weighted volume depends on the complete flags of the polytope. This connection is obtained by introducing flag simplices, which translate between the metric and…

Let $\Gamma$ be an $N\times n$ random matrix with independent entries and such that in each row entries are i.i.d. Assume also that the entries are symmetric, have unit variances, and satisfy a small ball probabilistic estimate uniformly.…

In this paper we deal with problems concerning the volume of the convex hull of two "connecting" bodies. After a historical background we collect some results, methods and open problems, respectively.

We study the rigidity of the volume entropy for weighted word metrics on hyperbolic groups, building on a recent convexity result due to Cantrell-Tanaka. Using ideas from small cancellation theory, we give conditions under which a…

We address the problem of constructing elliptic polytopes in R^d, which are convex hulls of finitely many two-dimensional ellipses with a common center. Such sets arise in the study of spectral properties of matrices, asymptotics of long…

We present a probabilistic argument supporting the application of polar duality, as discussed in our previous work, to express the indeterminacy principle of quantum mechanics. Our approach combines the properties of the Mahler volume of a…

Permanents of random matrices with independent and identically distributed (i.i.d.) entries have extensively studied in literature and convergence and concentration properties are known under varying assumptions on the distributions. In…

For a fixed $k\in\{1,\dots,d\}$ consider random vectors $X_0,\dots, X_{k}\in\mathbb R^d$ with an arbitrary spherically symmetric joint density function. Let $A$ be any non-singular $d\times d$ matrix. We show that the $k$-dimensional volume…

In this paper we give a formula for the probability that $n$ random points chosen under the uniform distribution in a disk are in convex position. While close, the formula is recursive and is totally explicit only for the first values of…

The volume is an important attribute of a convex body. In general, it is quite difficult to calculate the exact volume. But in many cases, it suffices to have an approximate value. Volume estimation methods for convex bodies have been…

We prove that for every convex body $K$ with the center of mass at the origin and every $\varepsilon\in \left(0,\frac{1}{2}\right)$, there exists a convex polytope $P$ with at most $e^{O(d)}\varepsilon^{-\frac{d-1}{2}}$ vertices such that…

We introduce the M-representation of polytopes, which makes it possible to compute linear transformations, convex hulls, and Minkowski sums with linear complexity in the dimension of the polytopes. When the polytope is a convex hull of a…

Let $K \subset \R^d$ be a smooth convex set and let $\P_\la$ be a Poisson point process on $\R^d$ of intensity $\la$. The convex hull of $\P_\la \cap K$ is a random convex polytope $K_\la$. As $\la \to \infty$, we show that the variance of…

We describe the harmonic interpolation of convex bodies, and prove a strong form of the Brunn-Minkowski inequality and characterize its equality case. As an application we improve a theorem of Berndtsson on the volume of slices of a…

In this paper we prove a theorem that provides an upper bound for the density of packings of congruent copies of a given convex body in $\mathbb{R}^n$; this theorem is a generalization of the linear programming bound for sphere packings. We…