Related papers: A note on the intersection property for flat boxes…

Given a point set $S$ in $\mathbb{R}^d$, a family of sets is $S$-intersecting if its members have a point in common in $S$. Recently, Edwards and Sober\'{o}n proved a fractional version of Halman's theorem for axis-parallel boxes, showing…

We discuss all possible compactifications on flat three-dimensional smooth spaces. In particular, various fields are studied on a box with opposite sides identified, after two of them are rotated by $\pi$, and their spectra are obtained.…

The main purpose of this paper is to study extremal results on the intersection graphs of boxes in $\R^d$. We calculate exactly the maximal number of intersecting pairs in a family $\F$ of $n$ boxes in $\R^d$ with the property that no $k+1$…

A hollow axis-aligned box is the boundary of the cartesian product of $d$ compact intervals in R^d. We show that for d\geq 3, if any 2^d of a collection of hollow axis-aligned boxes have non-empty intersection, then the whole collection has…

Let $\mathcal{F}$ be a family of $n$ axis-parallel boxes in $\mathbb{R}^d$ and $\alpha\in (1-1/d,1]$ a real number. There exists a real number $\beta(\alpha )>0$ such that if there are $\alpha {n\choose 2}$ intersecting pairs in…

We prove extensions of Halman's discrete Helly theorem for axis-parallel boxes in $\mathbb{R}^d$. Halman's theorem says that, given a set $S$ in $\mathbb{R}^d$, if $F$ is a finite family of axis-parallel boxes such that the intersection of…

Helly's theorem and its variants show that for a family of convex sets in Euclidean space, local intersection patterns influence global intersection patterns. A classical result of Eckhoff in 1988 provided an optimal fractional Helly…

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…

We remark that Pearl's Graphoid intersection property, also called intersection property in Bayesian networks, is a particular case of a general intersection property, in the sense of intersection of coverings, for factorisation spaces,…

We consider the moduli spaces $\mathcal{M}_d(\ell)$ of a closed linkage with n links and prescribed lengths in d-dimensional Euclidean space. For d>3 these spaces are no longer manifolds generically, but they have the structure of a…

Using geometric inversion with respect to the origin we extend the definition of box dimension to the case of unbounded subsets of Euclidean spaces. Alternative but equivalent definition is provided using stereographic projection on the…

We introduce a family of dimensions, which we call the $\Phi$-intermediate dimensions, that lie between the Hausdorff and box dimensions and generalise the intermediate dimensions introduced by Falconer, Fraser and Kempton. This is done by…

Let $X$ be a simplicial complex on vertex set $V$. We say that $X$ is $d$-representable if it is isomorphic to the nerve of a family of convex sets in $\mathbb{R}^d$. We define the $d$-boxicity of $X$ as the minimal $k$ such that $X$ can be…

We prove ``half-space" intersection properties in three settings: the hemisphere, half-geodesic balls in space forms, and certain subsets of Gaussian space. For instance, any two embedded minimal hypersurfaces in the sphere must intersect…

We introduce a new variant of quantitative Helly-type theorems: the minimal \emph{"homothetic distance"} of the intersection of a family of convex sets to the intersection of a subfamily of a fixed size. As an application, we establish the…

In this work we study the intersection properties of a finite disk system in the euclidean space. We accomplish this by utilizing subsets of spheres with varying dimensions and analyze specific points within them, referred to as poles.…

Let $A$ and $B$ be finite sets and consider a partition of the \emph{discrete box} $A \times B$ into \emph{sub-boxes} of the form $A' \times B'$ where $A' \subset A$ and $B' \subset B$. We say that such a partition has the…

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 present a polynomial partitioning theorem for finite sets of points in the real locus of an irreducible complex algebraic variety of codimension at most two. This result generalizes the polynomial partitioning theorem on the Euclidean…

We study families of axis-aligned boxes in a $d$-dimensional Euclidean space $\mathbb{R}^d$ whose placement is restricted by bounds on the dimension of their pairwise intersections. More specifically, two such boxes in $\mathbb{R}^d$ are…