Related papers: On the hollow enclosed by convex sets

Let $d \in \N$ and let $\D^d$ denote the class of all pairs $(R,M)$ in which $R = \bigoplus_{n \in \N_0} R_n$ is a Noetherian homogeneous ring with Artinian base ring $R_0$ and such that $M$ is a finitely generated graded $R$-module of…

Let $\mathcal F$ be a family of compact convex sets in $\mathbb R^d$. We say that $\mathcal F $ has a \emph{topological $\rho$-transversal of index $(m,k)$} ($\rho<m$, $0<k\leq d-m$) if there are, homologically, as many transversal…

We prove the existence (and give a characterization) of a germ of complex analytic set left invariant by an abelian group of germs of holomorphic diffeomorphisms at a common fixed point.We also give condition that ensure that such a group…

We study two combinatorial parameters, which we denote by f(S) and h(S), associated to an arbitrary set S \subseteq R^d, where d \in N. In the nondegenerate situation, f(S) is the largest possible number of facets of a d-dimensional…

Given a union-closed family $\mathcal{F}$ of subsets of the universe $[n]$, with $\mathcal{F}$ not equal to the power set of $[n]$, a new subset $A$ can be added to it such that the resulting family remains union-closed. We construct a new…

Let $P$ be a closed convex cone in $\mathbb{R}^{n}$. Assume that $P$ is spanning i.e. $P-P=\mathbb{R}^{n}$ and pointed i.e. $P \cap -P=\{0\}$. Let $\alpha:=\{\alpha_{x}:x \in P\}$ be a $\sigma$-weakly continuous family of unital normal…

Let $K\subseteq{\mathbb R}^n$ be a convex semialgebraic set. The semidefinite extension degree ${\mathrm{sxdeg}}(K)$ of $K$ is the smallest number $d$ such that $K$ is a linear image of an intersection of finitely many spectrahedra, each of…

The invisibility graph $I(X)$ of a set $X \subseteq \mathbb{R}^d$ is a (possibly infinite) graph whose vertices are the points of $X$ and two vertices are connected by an edge if and only if the straight-line segment connecting the two…

Some geometric facts concerning sets with positive reach in the Euclidean $d$-dimensional space ${\bf E}^d$ are proved. For $x_1$ and $x_2$ in ${\bf E}^d$ and $R>0$ let us denote by ${\mathfrak H}(x_1,x_2,R)$ the intersection of all closed…

A {\em convex hole} (or {\em empty convex polygon)} of a point set $P$ in the plane is a convex polygon with vertices in $P$, containing no points of $P$ in its interior. Let $R$ be a bounded convex region in the plane. We show that the…

Let $D$ be a non-pseudoconvex open set in $\C^3$ and $S$ be the union of all two-dimensional planes with non-empty and non-pseudoconvex intersection with $D.$ Sufficient conditions are given for $\C^3\setminus S$ to belong to a complex…

We consider the set of points chosen randomly, independently and uniformly in the $d$-dimensional spherical layer. A set of points is called $1$-convex if all its points are vertices of the convex hull of this set. In \cite{3} an estimate…

The Union Closed Sets Conjecture states that in every finite, nontrivial set family closed under taking unions there is an element contained in at least half of all the sets of the family. We investigate two new directions with respect to…

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…

Suppose that $\mathscr{F}$ is a finite union-closed family of sets with $\cup_{A\in \mathscr{F}}A=\{1,2,\ldots,m\}$ and $m\geq 2$. Fix $i\in \{1,2,\ldots,m\}$ and denote $\mathscr{G}:=\{A\backslash \{i\}: A\in \mathscr{F}\}$. For $j\in…

Cover-free families are set systems used as solutions for a large variety of problems, and in particular, problems where we deal with $n$ elements and want to identify $d$ invalid ones among them by performing only $t$ tests ($t \leq n$).…

A family $F$ of sets is said to satisfy the $(p,q)$-property if among any $p$ sets of $F$ some $q$ intersect. The celebrated $(p,q)$-theorem of Alon and Kleitman asserts that any family of compact convex sets in $\mathbb{R}^d$ that…

Our aim in this note is to show that, for any $\epsilon>0$, there exists a union-closed family $\mathcal F$ with (unique) smallest set $S$ such that no element of $S$ belongs to more than a fraction $\epsilon$ of the sets in $\mathcal F$.…

Let $n$ be any positive integer and $\mathcal{F}$ be a family of subsets of $[n]$. A family $\mathcal{F}'$ is said to be $D$-\emph{secting} for $\mathcal{F}$ if for every $A \in \mathcal{F}$, there exists a subset $A' \in \mathcal{F}'$ such…

A matrix convex set is a set of the form $\mathcal{S} = \cup_{n\geq 1}\mathcal{S}_n$ (where each $\mathcal{S}_n$ is a set of $d$-tuples of $n \times n$ matrices) that is invariant under UCP maps from $M_n$ to $M_k$ and under formation of…