Related papers: On the hollow enclosed by convex sets

The paper is concerned with families of plane algebraic curves that contain a given and quite special finite set X of points in the projective plane. We focus on the case in which the set X is formed by transversally intersecting pairs of…

In the convex covering problem, we are given a convex polygon with holes $P$ and the goal is to cover $P$ using a small number of convex polygons that lie inside $P$. In this paper, we solve the problem using the following strategy. We find…

Let $D$ be an orientation of a simple graph. Given $u,v\in V(D)$, a directed shortest $(u,v)$-path is a $(u,v)$-geodesic. $S \subseteq V(D)$ is convex if, for every $u,v \in S$, the vertices in each $(u,v)$-geodesic and in each…

The Frankl's conjecture, formulated in 1979. and still open, states that in every family of sets closed for unions there is an element contained in at least half of the sets. A family Fc is called Frankl-complete (or FC-family) if in every…

The Galvin problem asks for the minimum size of a family $\mathcal{F} \subseteq \binom{[n]}{n/2}$ with the property that, for any set $A$ of size $\frac n 2$, there is a set $S \in \mathcal{F}$ which is balanced on $A$, meaning that $|S…

The simplest way to generate a lattice of convex sets is to consider an initial set of points and draw segments, triangles, and any convex hull from it, then intersect them to obtain new points, and so forth. The result is an infinite…

A family of sets F is said to be union-closed if A \cup B is in F for every A and B in F. Frankl's conjecture states that given any finite union-closed family of sets, not all empty, there exists an element contained in at least half of the…

The closure of a discrete exponential family is described by a finite set of equations corresponding to the circuits of an underlying oriented matroid. These equations are similar to the equations used in algebraic statistics, although they…

Let $3\le d\le k$ and $\nu\ge 0$ be fixed and $\mathcal{F}\subset\binom{[n]}{k}$. The matching number of $\mathcal{F}$, denoted by $\nu(\mathcal{F})$, is the maximum number of pairwise disjoint sets in $\mathcal{F}$, and $\mathcal{F}$ is…

We introduce a generalization of the McMullen family $f_{\lambda}(z)=z^n+\lambda/z^d$. In 1988, C. McMullen showed that the Julia set of $f_{\lambda}$ is a Cantor set of circles if and only if $1/n+1/d<1$ and the simple critical values of…

We give a proof to the following theorem, which is well-known among experts: A connected subcomplex $W$ of a finite dimensional CAT(0) cubed complex $X$ is convex if and only if Lk$(v, W)$ is a full subcomplex of Lk$(v, X)$ for every vertex…

The union-closed sets conjecture (Frankl's conjecture) says that for any finite union-closed family of finite sets, other than the family consisting only of the empty set, there exists an element that belongs to at least half of the sets in…

The task of this survey is to present various results on intersection patterns of convex sets. One of main tools for studying intersection patterns is a point of view via simplicial complexes. We recall the definitions of so called…

For certain families of compact subsets of the plane, the isomorphism class of the algebra of absolutely continuous functions on a set is completely determined by the homeomorphism class of the set. This is analogous to the…

For a finite point set $P \subset \mathbb{R}^d$, denote by $\text{diam}(P)$ the ratio of the largest to the smallest distances between pairs of points in $P$. Let $c_{d, \alpha}(n)$ be the largest integer $c$ such that any $n$-point set $P…

Let $N$ be a finite set and $\mathcal{F}$, an intersection-closed family of subsets. Frankl conjectured that there always exists an element in $N$ which is contained in at most half the number of sets in $\mathcal{F}$ unless $\mathcal{F}…

The $K$-hull of a compact set $A\subset\mathbb{R}^d$, where $K\subset \mathbb{R}^d$ is a fixed compact convex body, is the intersection of all translates of $K$ that contain $A$. A set is called $K$-strongly convex if it coincides with its…

A spherical set is called convex if for every pair of its points there is at least one minimal geodesic segment that joins these points and lies in the set. We prove that for n >= 3 a complete locally-convex (topological) immersion of a…

The colourful simplicial depth conjecture states that any point in the convex hull of each of d+1 sets, or colours, of d+1 points in general position in R^d is contained in at least d^2+1 simplices with one vertex from each set. We verify…

We lay the foundations of convex hypersurface theory in contact topology, extending the work of Giroux in dimension three. Specifically, we prove that any closed hypersurface in a contact manifold can be $C^0$-approximated by a convex one.…