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Imre Bárány

Publications and source records attributed to Imre Bárány.

At least 37 records · Page 2Linked to original sources
arXiv2018

Pach's selection theorem does not admit a topological extension

Let $U_1,\dots, U_{d+1}$ be $n$-element sets in $R^d$ and let $\langle u_1,\ldots,u_{d+1}\rangle$ denote the convex hull of points $u_i$ in $U_i$ (for all $i$) which is a (possibly degenerate) simplex. Pach's selection theorem says that there are sets $Z_1 \subset U_1,\dots, Z_{d+1} \subset U_{d+1}$ and a point $u$ in $R^d$ such that each $|Z_i| > c_1(d)n$ and $u$ belongs to $\langle z_1,...,z_{d+1} \rangle$ for every choice of $z_1$ in $Z_1,\dots,z_{d+1}$ in $Z_{d+1}$. Here we show that this theorem does not admit a topological extension with linear size sets $Z_i$. However, there is a topological extension where each $|Z_i|$ is of order $(\log n)^(1/d)$.

math.CO↗
arXiv2018

Tverberg's theorem is 50 years old: a survey

This survey presents an overview of the advances around Tverberg's theorem, focusing on the last two decades. We discuss the topological, linear-algebraic, and combinatorial aspects of Tverberg's theorem and its applications. The survey contains several open problems and conjectures.

math.CO↗
arXiv2018

Convex cones, integral zonotopes, limit shape

This paper is about integral zonotopes. It is proven that large zonotopes in a convex cone have a limit shape, meaning that, after suitable scaling, the overwhelming majority of the zonotopes are very close to a fixed convex set. Several combinatorial properties of large zonotopes are established.

math.CO↗
arXiv2017

Tverberg plus minus

We prove a Tverberg type theorem: Given a set $A \subset \mathbb{R}^d$ in general position with $|A|=(r-1)(d+1)+1$ and $k\in \{0,1,\ldots,r-1\}$, there is a partition of $A$ into $r$ sets $A_1,\ldots,A_r$ with the following property. The unique $z \in \bigcap_1^r \mathrm{aff} A_j$ can be written as an affine combination of the element in $A_j$: $z = \sum_{x \in A_j} α(x)x$ for every $p$ and exactly $k$ of the coefficients $α(x)$ are negative. The case $k=0$ is Tverberg's classical theorem.

math.GT↗
arXiv2017

Intrinsic volumes and Gaussian polytopes: the missing piece of the jigsaw

The intrinsic volumes of Gaussian polytopes are considered. A lower variance bound for these quantities is proved, showing that, under suitable normalization, the variances converge to strictly positive limits. The implications of this missing piece of the jigsaw in the theory of Gaussian polytopes are discussed.

math.PR↗
arXiv2017

Algebraic vertices of non-convex polyhedra

In this article we define an algebraic vertex of a generalized polyhedron and show that it is the smallest set of points needed to define the polyhedron. We prove that the indicator function of a generalized polytope $P$ is a linear combination of indicator functions of simplices whose vertices are algebraic vertices of $P$. We also show that the indicator function of any generalized polyhedron is a linear combination, with integer coefficients, of indicator functions of cones with apices at algebraic vertices and line-cones. The concept of an algebraic vertex is closely related to the Fourier--Laplace transform. We show that a point $\mathbf{v}$ is an algebraic vertex of a generalized polyhedron $P$ if and only if the tangent cone of $P$, at $\mathbf{v}$, has non-zero Fourier--Laplace transform.

math.MG↗
arXiv2016

A Tverberg type theorem for matroids

Let b(M) denote the maximal number of disjoint bases in a matroid M. It is shown that if M is a matroid of rank d+1, then for any continuous map f from the matroidal complex M into the d-dimensional Euclidean space there exist t \geq \sqrt{b(M)}/4 disjoint independent sets σ_1,\ldots,σ_t \in M such that \bigcap_{i=1}^t f(σ_i) \neq \emptyset.

math.CO↗
arXiv2016

Gershgorin disks for multiple eigenvalues of non-negative matrices

Gershgorin's famous circle theorem states that all eigenvalues of a square matrix lie in disks (called Gershgorin disks) around the diagonal elements. Here we show that if the matrix entries are non-negative and an eigenvalue has geometric multiplicity at least two, then this eigenvalue lies in a smaller disk. The proof uses geometric rearrangement inequalities on sums of higher dimensional real vectors which is another new result of this paper.

math.CO↗
arXiv2015

Random points in halfspheres

A random spherical polytope $P_n$ in a spherically convex set $K \subset S^d$ as considered here is the spherical convex hull of $n$ independent, uniformly distributed random points in $K$. The behaviour of $P_n$ for a spherically convex set $K$ contained in an open halfsphere is quite similar to that of a similarly generated random convex polytope in a Euclidean space, but the case when $K$ is a halfsphere is different. This is what we investigate here, establishing the asymptotic behaviour, as $n$ tends to infinity, of the expectation of several characteristics of $P_n$, such as facet and vertex number, volume and surface area. For the Hausdorff distance from the halfsphere, we obtain also some almost sure asymptotic estimates.

math.PR↗
arXiv2015

Erdős - Szekeres Theorem for Lines

According to the Erdős-Szekeres theorem, for every $n$, a sufficiently large set of points in general position in the plane contains $n$ in convex position. In this note we investigate the line version of this result, that is, we want to find $n$ lines in convex position in a sufficiently large set of lines that are in general position. We prove almost matching upper and lower bounds for the minimum size of the set of lines in general position that always contains $n$ in convex position. This is quite unexpected, since in the case of points, the best known bounds are very far from each other. We also establish the dual versions of many variants and generalizations of the Erd\H os-Szekeres theorem.

math.MG↗
arXiv2014

$(n,m)$-Fold Covers of Spheres

A well known consequence of the Borsuk-Ulam theorem is that if the $d$-dimensional sphere $S^d$ is covered with less than $d+2$ open sets, then there is a set containing a pair of antipodal points. In this paper we provide lower and upper bounds on the minimum number of open sets, not containing a pair of antipodal points, needed to cover the $d$-dimensional sphere $n$ times, with the additional property that the northern hemisphere is covered $m > n$ times. We prove that if the open northern hemisphere is to be covered $m$ times then at least $ \lceil \frac{d-1}{2} \rceil+n+m$ and at most $d+n+m$ sets are needed. For the case of $n=1$ and $d \ge 2$, this number is equal to $d+2$ if $m \le \lfloor \frac{d}{2} \rfloor + 1$ and equal to $ \lfloor \frac{d-1}{2} \rfloor + 2 +m$ if $m > \lfloor \frac{d}{2} \rfloor + 1$. If the closed northern hemisphere is to be covered $m$ times then $d+2m-1$ sets are needed, this number is also sufficient. We also present results on a related problem of independent interest. We prove that if $S^d$ is covered $n$ times with open sets, not containing a pair of antipodal points, then there exists a point that is covered at least $ \lceil \frac{d}{2} \rceil +n$ times. Furthermore, we show that there are covers in which no point is covered more than $n+d$ times.

math.CO↗
arXiv2014

Block Partitions of Sequences

Given a sequence A=(a1,...,an) of real numbers, a block B of the A is either a set B={ai,...,aj} where i<=j or the empty set. The size b of a block B is the sum of its elements. We show that when 0<=ai<=1 and k is a positive integer, there is a partition of A into k blocks B1,...,Bk with |bi-bj|<=1 for every i, j. We extend this result in many directions.

math.CO↗
arXiv2014

Magic mirrors, dense diameters, Baire category

An old result of Zamfirescu says that for most convex curves $C$ in the plane most points in $R^2$ lie on infinitely many normals to $C$, where most is meant in Baire category sense. We strengthen this result by showing that `infinitely many' can be replaced by `contiunuum many' in the statement. We present further theorems in the same spirit.

math.MG↗
arXiv2012

Many empty triangles have a common edge

Given a finite point set $X$ in the plane, the degree of a pair $\{x,y\} \subset X$ is the number of empty triangles $t=conv\{x,y,z\}$, where empty means $t\cap X=\{x,y,z\}$. Define $deg X$ as the maximal degree of a pair in $X$. Our main result is that if $X$ is a random sample of $n$ independent and uniform points from a fixed convex body, then $deg X \ge cn/\ln n$ in expectation.

math.PR↗
arXiv2010

Poisson polytopes

We prove the central limit theorem for the volume and the $f$-vector of the Poisson random polytope $Π_η$ in a fixed convex polytope $P\subset\mathbb{R}^d$. Here, $Π_η$ is the convex hull of the intersection of a Poisson process $X$ of intensity $η$ with $P$.

math.PR↗
arXiv2009

Intrinsic volumes of inscribed random polytopes in smooth convex bodies

Let $K$ be a $d$ dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by $K_n$ the convex hull of $n$ points chosen randomly and independently from $K$ according to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of the $s$-th intrinsic volumes $V_s(K_n)$ of $K_n$ for $s\in\{1, ..., d\}$. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of $K_n$. The essential tools are the Economic Cap Covering Theorem of Bárány and Larman, and the Efron-Stein jackknife inequality.

math.MG↗
arXiv2004

Balanced Partitions of Vector Sequences

Let $d, r \in \N$, $\|\cdot\|$ any norm on $\R^d$ and $B$ denote the unit ball with respect to this norm. We show that any sequence $v_1,v_2,...$ of vectors in $B$ can be partitioned into $r$ subsequences $V_1, ..., V_r$ in a balanced manner with respect to the partial sums: For all $n \in \N$, $\ell \le r$, we have $\|\sum_{i \le k, v_i \in V_\ell} v_i - \tfrac 1r \sum_{i \le k} v_i\| \le 2.0005 d$. A similar bound holds for partitioning sequences of vector sets. Both results extend an earlier one of Bárány and Grinberg (1981) to partitions in arbitrarily many classes.

math.CO↗