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Imre Barany

Publications and source records attributed to Imre Barany.

At least 19 recordsLinked to original sources
arXiv2025

Excursions in Sylvester-Gallai land

The Sylvester-Gallai theorem states that for a finite set of points in the plane, if every line determined by any two of these points also contains a third, then the set is necessarily made of collinear points. In this paper, we first provide a counterexample in the plane when the point set is countably infinite but bounded. Then we consider a variant of the Sylvester-Gallai theorem where instead of a finite point set we have a finite family of convex sets in $\mathbb{R}^d$ ($d\geq 2$). Finally, we present another variant of the Sylvester-Gallai theorem, when instead of point sets we have a finite family of line-segments in the plane.

math.CO↗
arXiv2024

A matrix version of the Steinitz lemma

The Steinitz lemma, a classic from 1913, states that $a_1,\ldots,a_n$, a sequence of vectors in $\R^d$ with $\sum_1^n a_i=0$, can be rearranged so that every partial sum of the rearranged sequence has norm at most $2d\max \|a_i\|$. In the matrix version $A$ is a $k\times n$ matrix with entries $a_i^j \in \R^d$ with $\sum_{j=1}^k\sum_{i=1}^na_i^j=0$. It is proved in \cite{OPW} that there is a rearrangement of row $j$ of $A$ (for every $j$) such that the sum of the entries in the first $m$ columns of the rearranged matrix has norm at most $40d^5\max \|a_i^j\|$ (for every $m$). We improve this bound to $(4d-2)\max \|a_i^j\|$.

math.CO↗
arXiv2023

Tverberg's theorem, a new proof

We give a new proof Tverberg's famous theorem: For every set $X \subset \R^d$ with $|X|=(r-1)(d+1)+1$, there is a partition of $X$ into $r$ sets $X_1,\ldots,X_r$ such that $\bigcap_{p=1}^r \conv X_p\ne \emptyset$. The new proof uses linear algebra, specially structured matrices, the theory of linear equations, and Tverberg's original ``moving the points" method.

math.CO↗
arXiv2023

Positive bases, cones, Helly type theorems

Assume that $k \le d$ is a positive integer and $\C$ is a finite collection of convex bodies in $\R^d$. We prove a Helly type theorem: If for every subfamily $\C^*\subset \C$ of size at most $\max \{d+1,2(d-k+1)\}$ the set $\bigcap \C^*$ contains a $k$-dimensional cone, then so does $\bigcap \C.$ One ingredient in the proof is another Helly type theorem about the dimension of lineality spaces of convex cones.

math.MG↗
arXiv2022

Cells in the box and a hyperplane

It is well-known that a line can intersect at most $2n-1$ cells of the $n \times n$ chessboard. Here we consider the high dimensional version: how many cells of the $d$-dimensional $n\times \ldots \times n$ box can a hyperplane intersect? We also prove the lattice analogue of the following well-known fact. If $K,L$ are convex bodies in $R^d$ and $K\subset L$, then the surface area of $K$ is smaller than that of $L$.

math.MG↗
arXiv2021

Pairwise intersecting convex sets and cylinders in $\R^3$

We prove that given a finite collection of cylinders in $\R^3$ with the property that any two them intersect, then there is a line intersecting an $α$ fraction of the cylinders where $α=\frac 1{28}$. This is a special case of an interesting conjecture.

math.CO↗
arXiv2020

Small subset sums

Let ||.|| be a norm in R^d whose unit ball is B. Assume that V\subset B is a finite set of cardinality n, with \sum_{v \in V} v=0. We show that for every integer k with 0 \le k \le n, there exists a subset U of V consisting of k elements such that \| \sum_{v \in U} v \| \le \lceil d/2 \rceil. We also prove that this bound is sharp in general. We improve the estimate to O(\sqrt d) for the Euclidean and the max norms. An application on vector sums in the plane is also given.

math.MG↗
arXiv2015

Topology of geometric joins

We consider the geometric join of a family of subsets of the Euclidean space. This is a construction frequently used in the (colorful) Carathéodory and Tverberg theorems, and their relatives. We conjecture that when the family has at least $d+1$ sets, where $d$ is the dimension of the space, then the geometric join is contractible. We are able to prove this when $d$ equals $2$ and $3$, while for larger $d$ we show that the geometric join is contractible provided the number of sets is quadratic in $d$. We also consider a matroid generalization of geometric joins and provide similar bounds in this case.

math.MG↗
arXiv2014

Affine diameters of convex bodies

We prove sharp inequalities for the average number of affine diameters through the points of a convex body $K$ in ${\mathbb R}^n$. These inequalities hold if $K$ is either a polytope or of dimension two. An example shows that the proof given in the latter case does not extend to higher dimensions.

math.MG↗
arXiv2014

Typical curvature behaviour of bodies of constant width

It is known that an $n$-dimensional convex body which is typical in the sense of Baire category, shows a simple, but highly non-intuitive curvature behaviour: at almost all of its boundary points, in the sense of measure, all curvatures are zero, but there is also a dense and uncountable set of boundary points at which all curvatures are infinite. The purpose of this paper is to find a counterpart to this phenomenon for typical convex bodies of given constant width. Such bodies cannot have zero curvatures. A main result says that for a typical $n$-dimensional convex body of constant width $1$ (without loss of generality), at almost all boundary points, in the sense of measure, all curvatures are equal to $1$. (In contrast, note that a ball of width $1$ has radius $1/2$, hence all its curvatures are equal to $2$.) Since the property of constant width is linear with respect to Minkowski addition, the proof requires recourse to a linear curvature notion, which is provided by the tangential radii of curvature.

math.MG↗
arXiv2014

Volumes of convex lattice polytopes and a question of V. I. Arnold

We show by a direct construction that there are at least $\exp\{cV^{(d-1)/(d+1)}\}$ convex lattice polytopes in $\mathbb{R}^d$ of volume $V$ that are different in the sense that none of them can be carried to an other one by a lattice preserving affine transformation. This is achieved by considering the family $\mathcal{P}^d(r)$ (to be defined in the text) of convex lattice polytopes whose volumes are between $0$ and $r^d/d!$. Namely we prove that for $P \in \mathcal{P}^d(r)$, $d!\mathrm{vol\;} P$ takes all possible integer values between $cr^{d-1}$ and $r^d$ where $c>0$ is a constant depending only on $d$.

math.CO↗
arXiv2013

Curves in R^d intersecting every hyperplane at most d+1 times

By a curve in R^d we mean a continuous map gamma:I -> R^d, where I is a closed interval. We call a curve gamma in R^d at most k crossing if it intersects every hyperplane at most k times (counted with multiplicity). The at most d crossing curves in R^d are often called convex curves and they form an important class; a primary example is the moment curve {(t,t^2,...,t^d):t\in[0,1]}. They are also closely related to Chebyshev systems, which is a notion of considerable importance, e.g., in approximation theory. We prove that for every d there is M=M(d) such that every at most d+1 crossing curve in R^d can be subdivided into at most M convex curves. As a consequence, based on the work of Elias, Roldan, Safernova, and the second author, we obtain an essentially tight lower bound for a geometric Ramsey-type problem in R^d concerning order-type homogeneous sequences of points, investigated in several previous papers.

math.MG↗
arXiv2013

2013 Unit Vectors in the Plane

Given a norm on the plane and 2013 unit vectors in this norm, there is a signed sum of these vectors whose norm is at most one.

math.CO↗
arXiv2011

Notes about the Caratheodory number

In this paper we give sufficient conditions for a compactum in $\mathbb R^n$ to have Carathéodory number less than $n+1$, generalizing an old result of Fenchel. Then we prove the corresponding versions of the colorful Carathéodory theorem and give a Tverberg type theorem for families of convex compacta.

math.MG↗
arXiv2010

Slicing convex sets and measures by a hyperplane

We generalize the ham sandwich theorem for the case of well separated measures. Given convex bodies $K_1,...,K_d$ in $\mathbb{R_d}$ and numbers $α_1,...,α_d \in [0, 1]$, we give a sufficient condition for existence and uniqueness of an (oriented)halfspace H with Vol($H \cap K_i$)= $α_i \dot$ Vol$K_i$ for every i. The result is extended from convex bodies to measures.

math.CO↗
arXiv2009

Longest convex chains

Assume $X_n$ is a random sample of $n$ uniform, independent points from a triangle $T$. The longest convex chain, $Y$, of $X_n$ is defined naturally. The length $|Y|$ of $Y$ is a random variable, denoted by $L_n$. In this article, we determine the order of magnitude of the expectation of $L_n$. We show further that $L_n$ is highly concentrated around its mean, and that the longest convex chains have a limit shape.

math.PR↗
arXiv2006

Strictly convex drawings of planar graphs

Every three-connected planar graph with n vertices has a drawing on an O(n^2) x O(n^2) grid in which all faces are strictly convex polygons. These drawings are obtained by perturbing (not strictly) convex drawings on O(n) x O(n) grids. More generally, a strictly convex drawing exists on a grid of size O(W) x O(n^4/W), for any choice of a parameter W in the range n<W<n^2. Tighter bounds are obtained when the faces have fewer sides. In the proof, we derive an explicit lower bound on the number of primitive vectors in a triangle.

cs.CG↗