Related papers: The intersection of typical Besicovitch sets with …

For every nonempty compact convex subset $K$ of a normed linear space a (unique) point $c_K \in K$, called the generalized Chebyshev center, is distinguished. It is shown that $c_K$ is a common fixed point for the isometry group of the…

We consider shifts of a set $A\subseteq\mathbb{N}$ by elements from another set $B\subseteq\mathbb{N}$, and prove intersection properties according to the relative asymptotic size of $A$ and $B$. A consequence of our main theorem is the…

In [8] we found a class of overlapping asymmetric self-similar measures on the real line, which are generically absolutely continuous with respect to the Lebesgue measure. Here we construct exceptional measures in this class being singular.

For a metrizable space $X$, we denote by $\mathrm{Met}(X)$ the space of all metric that generate the same topology of $X$. The space $\mathrm{Met}(X)$ is equipped with the supremum distance. In this paper, for every strongly…

In this paper we focus on minimal Besicovitch arrangements to highlight some of their properties. An appropriate probability space enables us to find again in an elegant way some straightforward equalities associated with these…

Huemer et al. (Discrete Math, 2019) proved that for any two finite point sets $R$ and $B$ in the plane with $|R| = |B|$, the perfect matching that matches points of $R$ with points of $B$, and maximizes the total squared Euclidean distance…

We describe general connections between intersective properties of sets in Abelian groups and positive exponential sums. In particular, given a set $A$ the maximal size of a set whose difference set avoids $A$ will be related to positive…

We pose the following conjecture: (*) If A is the union of line segments in R^n, and B is the union of the corresponding full lines then the Hausdorff dimensions of A and B agree. We prove that this conjecture would imply that every…

Several authors have recently attempted to show that the intersection of three simply connected subcontinua of the plane is simply connected provided it is non-empty and the intersection of each two of the continua is path connected. In…

Let B be a finite collection of geometric (not necessarily convex) bodies in the plane. Clearly, this class of geometric objects naturally generalizes the class of disks, lines, ellipsoids, and even convex polygons. We consider geometric…

The class of intersection bigraphs of unit intervals of the real line whose ends may be open or closed is called a class of mixed unit interval bigraphs. This class of bigraphs is a strict superclass of the class of unit interval bigraphs.…

In this paper, we provide a combinatorial characterization of those collections of cells whose inner $2$-minor ideals are complete intersections. More precisely, given a collection of cells $\mathcal C$ and its associated inner $2$-minor…

A family of sets is said to be intersecting if every pair of sets in the family have non-empty intersection. In this paper, we initiate the study of intersecting non-uniform families of sets of one of two sizes containing given subfamilies.…

The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper radical ideals (e.g., $\{0\})$ that are closed under the natural metric, but has no prime ideals closed under that metric; hence closed…

Let $\mathcal G$ be a family of subsets of an $n$-element set. The family $\mathcal G$ is called non-trivial $3$-wise intersecting if the intersection of any three subsets in $\mathcal G$ is non-empty, but the intersection of all subsets is…

The goal of this paper is to prove the theorem in the title.

A totally symmetric set is a finite subset of a group for which any permutation of the elements can be realized by conjugation in the ambient group. Such sets are rigid under homomorphisms, and so exert a great deal of control over the…

Let $\mathcal A=\{A_1,\ldots,A_n\}$ be a family of sets in the plane. For $0 \leq i < n$, denote by $f_i$ the number of subsets $\sigma$ of $\{1,\ldots,n\}$ of cardinality $i+1$ that satisfy $\bigcap_{i \in \sigma} A_i \neq \emptyset$. Let…

We prove that in the Hilbert space every uniformly convex set with modulus of convexity of the second order at zero is an intersection of closed balls of fixed radius. We also obtain an estimate of this radius.

The aim of this paper is to introduce the class of ${\cal A}{\cal B}$-sets as the sets that are the intersection of an open and a semi-regular set. Several classes of well-known topological spaces are characterized via the new concept. A…