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Imre Bárány

Publications and source records attributed to Imre Bárány.

At least 19 recordsLinked to original sources
arXiv2026

Line arrangements, bounded cells, and discrepancy

An arrangement of $n$ lines in the plane defines cells, i.e., connected components of the complement of the arrangement. Answering a question of Volkmar Welker we show that, apart from two special cases, there is always a line $L$ in the arrangement such that one of the open halfplanes whose boundary is $L$ contains more than half of the bounded cells of the arrangement. We also consider some unusual discrepancy problems that are related to this question.

math.CO↗
arXiv2026

The Volume Helly Theorem in the plane, colorful version

We prove a colorful volume Helly theorem for convex sets in $\mathbb R^2$: There is a constant $V>0$ such that if $\mathfrak F_1,\mathfrak F_2,\mathfrak F_3,\mathfrak F_4$ are finite families of convex sets in $\mathbb R^2$ and if $|\bigcap_1^4F_i|\ge V$ for every transversal $F_i\in \mathfrak F_i,\; (i=1,\ldots,4)$, then $|\bigcap \mathfrak{F}_i|\ge 1$ for some $i$. Here $|A|$ is the Lebesgue measure of $A\subset \mathbb R^d$. The main ingredient is the following theorem. Let $Q_1,\ldots,Q_4\subset\mathbb R^2$ be convex quadrilaterals of area at most $1$, where of course each $Q_i$ is the intersection of 4 halfplanes. Then for every $Q_i$ there is one of these halfplanes $H_i$, say, such that $|\bigcap_1^4 H_i| \le 4096$.

math.CO↗
arXiv2026

A point in the interior of the convex hulls

Steinitz's theorem states that if a point $a \in \mathrm{int\,conv\,} X$ for a set $X \subset \mathbb{R}^d$, then $X$ contains a subset $Y$ of size at most $2d$ such that $a \in \mathrm{int\,conv\,}Y$. The bound $2d$ is best possible here. We prove the colourful version of this theorem and characterize the cases when exactly $2d$ sets are needed.

math.CO↗
arXiv2026

The integer hull of the set $\{(x,y)\in \mathbb{R}^2: xy\ge N\}$

The integer convex hull $I(H_N)$ of the set $H_N=\{(x,y)\in \mathbb{R}^2: xy\ge N\}$ is the convex hull of the lattice points in $H_N$. The vertices of $I(H_N)$ lie in the square $[1,N]^2$. Improving on a recent result of Alcántara et al. ~\cite{Santos} we show that the number of vertices of $I(H_N)$ is of order $N^{1/3}\log N$. We also show that the area of the part of $H_N \setminus I(H_N)$ that lies in the square $[1,N^{2/3}]^2$ is also of order $N^{1/3}\log N$.

math.CO↗
arXiv2025

Balancing games on unbounded sets

For a finite set $V\subset \mathbb{R}^n$, a set $T\subset \mathbb{R}^n$ is called $V$-closed if $t \in T$ and $v\in V$ imply that either $t+v\in T$ or $t-v \in T$. The set $P(V):=\{\sum_{v \in W} v: W \subset V\}$ is clearly $V$-closed and so are its translates. We show, assuming $V$ contains no parallel vectors, that if $T$ is closed and $V$-closed, and $x \in T$ is an extreme point of $\operatorname{cl} \operatorname{conv} T$, then there is a translate of $P(V)$ containing $x$ and contained in $\operatorname{conv} T$. This result is used to determine the value of a special balancing game. A byproduct is that when $m\ge 2$ and is not a power of 2, then the $m$-sets of a $2m$-set can be coloured Red and Blue so that complementary $m$-sets have distinct colours and every point of the $2m$-set is contained in the same number of Red and Blue sets.

math.CO↗
arXiv2025

Piercing intersecting convex sets

Assume two finite families $\mathcal A$ and $\mathcal B$ of convex sets in $\mathbb{R}^3$ have the property that $A\cap B\ne \emptyset$ for every $A \in \mathcal A$ and $B\in \mathcal B$. Is there a constant $γ>0$ (independent of $\mathcal A$ and $\mathcal B$) such that there is a line intersecting $γ|\mathcal A|$ sets in $\mathcal A$ or $γ|\mathcal B|$ sets in $\mathcal B$? This is an intriguing Helly-type question from a paper by Martínez, Roldan and Rubin. We confirm this in the special case when all sets in $\mathcal A$ lie in parallel planes and all sets in $\mathcal B$ lie in parallel planes; in fact, all sets from one of the two families has a line transversal.

math.CO↗
arXiv2024

A special balancing game, Victor Grinberg's question

In a vector balancing game, on step $k$ player 1 chooses a unit vector $v_k$ and player 2 chooses a sign $\varepsilon_k\in \{-1,1\}$, and the position after $n$ steps is $z_n=\sum_1^n\varepsilon_k v_k$. Player 1's target is to make $\|z_n\|$ large and player 2's target is to make it small. We consider a special case of this game.

math.CO↗
arXiv2024

Same average in every direction

Given a polytope $P\subset R^3$ and a non-zero vector $z \in R^3$, the plane $\{x\in R^3:zx=t\}$ intersects $P$ in convex polygon $P(z,t)$ for $t \in [t^-,t^+]$ where $t^-=\min \{zx: x \in P\}$ and $t^+=\max \{zx: x\in P\}$, $zx$ is the scalar product of $z,x \in R^3$. Let $A(P,z)$ denote the average number of vertices of $P(z,t)$ on the interval $[t^-,t^+]$. For what polytopes is $A(P,z)$ a constant independent of $z$?

math.CO↗
arXiv2023

Piercing the chessboard

We consider the minimum number of lines $h_n$ and $p_n$ needed to intersect or pierce, respectively, all the cells of the $n \times n$ chessboard. Determining these values can also be interpreted as a strengthening of the classical plank problem for integer points. Using the symmetric plank theorem of K. Ball, we prove that $h_n = \lceil \frac n 2 \rceil$ for each $n \geq 1$. Studying the piercing problem, we show that $0.7n \leq p_n \leq n-1$ for $n\geq 3$, where the upper bound is conjectured to be sharp. The lower bound is proven by using the linear programming method, whose limitations are also demonstrated.

math.CO↗
arXiv2022

Smaller Gershgorin disks for multiple eigenvalues for complex matrices

Extending an earlier result for real matrices we show that multiple eigenvalues of a complex matrix lie in a reduced Gershgorin disk. One consequence is a slightly better estimate in the real case. Another one is a geometric application. Further results of a similar type are given for normal and almost symmetric matrices.

math.RA↗
arXiv2021

Helly-type Problems

In this paper, we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carathéodory, and Tverberg. Through these problems we describe the fascinating area of Helly-type theorems, and explain some of its main themes and goals.

math.CO↗
arXiv2021

Universal sequences of lines in $\mathbb R^d$

One of the most important and useful examples in discrete geometry is a finite sequence of points on the moment curve $γ(t)=(t,t^2,t^3,\dots ,t^d)$ or, more generally, on a {\it strictly monotone curve} in $\mathbb R^d$. These sequences as well as the ambient curve itself can be described in terms of {\it universality properties} and we will study the question: "What is a universal sequence of oriented and unoriented lines in $d$-space'' We give partial answers to this question, and to the analogous one for $k$-flats. Given a large integer $n$, it turns out that, like the case of points the number of universal configurations is bounded by a function of $d$, but unlike the case for points, there are a large number of distinct universal finite sequences of lines. We show that their number is at least $2^{d-1}-2$ and at most $(d-1)!$. However, like for points, in all dimensions except $d=4$, there is essentially a unique {\em continuous} example of a universal family of lines. The case $d=4$ is left as an open question.

math.CO↗
arXiv2020

The cocked hat

We revisit the cocked hat -- an old problem from navigation -- and examine under what conditions its old solution is valid.

math.MG↗
arXiv2019

Theorems of Carathéodory, Helly, and Tverberg without dimension

We prove a no-dimensional version of Carathédory's theorem: given an $n$-element set $P\subset \Re^d$, a point $a \in \conv P$, and an integer $r\le d$, $r \le n$, there is a subset $Q\subset P$ of $r$ elements such that the distance between $a$ and $\conv Q$ is less than $\diam P/\sqrt {2r}$. A general no-dimension Helly type result is also proved with colourful and fractional consequences. Similar versions of Tverberg's theorem and some of their extensions are also established.

math.MG↗
arXiv2019

Primitive points in rational polygons

Let $\mathcal A$ be a star-shaped polygon in the plane, with rational vertices, containing the origin. The number of primitive lattice points in the dilate $t\mathcal A$ is asymptotically $\frac6{π^2}$ Area$(t\mathcal A)$ as $t\to \infty$. We show that the error term is both $Ω_\pm\big( t\sqrt{\log\log t} \big)$ and $O(t(\log t)^{2/3}(\log\log t)^{4/3})$. Both bounds extend (to the above class of polygons) known results for the isosceles right triangle, which appear in the literature as bounds for the error term in the summatory function for Euler's $ϕ(n)$.

math.NT↗
arXiv2019

Almost similar configurations

Let $h(n)$ denote the maximum number of triangles with angles between $59^\circ$ and $61^\circ$ in any $n$-element planar set. Our main result is an exact formula for $h(n)$. We also prove $h(n)= n^3/24+ O(n \log n)$ as $n\to \infty$. However, there are triangles $T$ and $n$-point sets $P$ showing that the number of $\varepsilon$-similar copies of $T$ in $P$ can exceed $n^3/15$ for any $\varepsilon>0$.

math.CO↗
arXiv2019

Random polytopes and the wet part for arbitrary probability distributions

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 the uniform distribution from a convex set [Bárány and Larman 1988]. The lower bound of Bárány and Larman continues to hold in the general setting, but the upper bound must be relaxed by a factor of $\log n$. We show by an example that this is tight.

math.PR↗