Related papers: Helly numbers for Quantitative Helly-type results
We prove existence of Helly numbers for crystals and for cut-and-project sets with convex windows. Also we show that for a two-dimensional crystal consisting of $k$ copies of a single lattice the Helly number does not exceed $k+6$.
We consider quantitative versions of Helly-type questions, that is, instead of finding a point in the intersection, we bound the volume of the intersection. Our first main geometric result is a quantitative version of the Fractional Helly…
A finite family $\mathcal F$ of convex sets is $k$-intersecting in $S \subseteq \mathbb{R}^d$ if the intersection of every subset of $k$ convex sets in $\mathcal F$ contains a point in $S$. The Helly number of $S$ is the minimum $k$, if it…
In 2008, Halman showed that for any finite set $P\subset \mathbb R^d$ and any finite family $\mathcal{B}$ of axis-parallel boxes in $\mathbb{R}^d$, if the intersection of $P$ and any subfamily $\mathcal{B}' \subseteq\mathcal{B}$ of size at…
The Helly number of a family of sets with empty intersection is the size of its largest inclusion-wise minimal sub-family with empty intersection. Let F be a finite family of open subsets of an arbitrary locally arc-wise connected…
We prove discrete Helly-type theorems for pseudohalfplanes, which extend recent results of Jensen, Joshi and Ray about halfplanes. Among others we show that given a family of pseudohalfplanes $\cal H$ and a set of points $P$, if every…
Given a set $S \subseteq \mathbb{R}^2$, define the \emph{Helly number of $S$}, denoted by $H(S)$, as the smallest positive integer $N$, if it exists, for which the following statement is true: for any finite family $\mathcal{F}$ of convex…
Let $M$ be a subset of $\mathbb{R}^k$. It is an important question in the theory of linear inequalities to estimate the minimal number $h=h(M)$ such that every system of linear inequalities which is infeasible over $M$ has a subsystem of at…
We present a unified approach to prove Helly-type theorems for monotone properties of boxes, such as having large volume or containing points from a given set. As a corollary, we obtain new proofs for several earlier results regarding…
A convex lattice set in $\mathbb{Z}^d$ is the intersection of a convex set in $\mathbb{R}^d$ and the integer lattice $\mathbb{Z}^d$. A well-known theorem of Doignon states that the Helly number of $d$-dimensional convex lattice sets equals…
We prove the following variant of Helly's classical theorem for Hamming balls with a bounded radius. For $n>t$ and any (finite or infinite) set $X$, if in a family of Hamming balls of radius $t$ in $X^n$, every subfamily of at most…
Two celebrated extensions of the classical Helly's theorem are the fractional Helly theorem and the colorful Helly theorem. Bulavka, Goodarzi, and Tancer recently established the optimal bound for the unified generalization of the…
We report on some recent progress regarding combinatorial properties in convexity spaces with a bounded Radon number. In particular, we discuss the relationship between the Radon number, the colorful and fractional Helly properties, weak…
Brazitikos' results on quantititative Helly-type theorems (for the volume and for the diameter) rely on the work of Srivastava on sparsification of John's decompositions. We change this technique by a stronger recent result due to Friedland…
We establish a theorem regarding the maximum size of an {\it{induced}} matching in the bipartite complement of the incidence graph of a set system $(X,\mathcal{F})$. We show that this quantity plus one provides an upper bound on the…
Families of translates and homothets of strictly convex curves are proven to possess Helly-type properties generalizing those of a circle. Weaker results are shown for arbitrary convex curves.
We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers $b$ and $d$ there exists an integer $h(b,d)$ such that the following…
B\'ar\'any, Katchalski and Pach proved the following quantitative form of Helly's theorem. If the intersection of a family of convex sets in $\mathbb{R}^d$ is of volume one, then the intersection of some subfamily of at most $2d$ members is…
We show quantitative versions of classic results in discrete geometry, where the size of a convex set is determined by some non-negative function. We give versions of this kind for the selection theorem of B\'ar\'any, the existence of weak…
We prove variations of Carath\'eodory's, Helly's and Tverberg's theorems where the sets involved are measured according to continuous functions such as the volume or diameter. Among our results, we present continuous quantitative versions…