Related papers: Point Sets with Small Integer Coordinates and with…

In this paper, we study the properties of the Fermat-Weber point for a set of fixed points, whose arrangement coincides with the vertices of a regular polygonal chain. A $k$-chain of a regular $n$-gon is the segment of the boundary of the…

A point set $S \subseteq \mathbb{R}^2$ is universal for a class $\cal G$ if every graph of ${\cal G}$ has a planar straight-line embedding on $S$. It is well-known that the integer grid is a quadratic-size universal point set for planar…

For a finite set $A\subset \mathbb{R}^d$, let $\Delta(A)$ denote the spread of $A$, which is the ratio of the maximum pairwise distance to the minimum pairwise distance. For a positive integer $n$, let $\gamma_d(n)$ denote the largest…

Tverberg's theorem states that any set of $t(r,d)=(r-1)(d+1)+1$ points in $\mathbb{R}^d$ can be partitioned into $r$ subsets whose convex hulls have non-empty $r$-fold intersection. Moreover, generic collections of fewer points cannot be so…

Let $h(n)$ be the minimum integer such that every complete $n$-vertex simple topological graph contains an edge that crosses at most $h(n)$ other edges. In 2009, Kyn\v{c}l and Valtr showed that $h(n) = O(n^2/\log^{1/4} n)$, and in the other…

A classic theorem of Euclidean geometry asserts that any noncollinear set of $n$ points in the plane determines at least $n$ distinct lines. Chen and Chv\'atal conjectured that this holds for an arbitrary finite metric space, with a certain…

A classical theorem of Spencer shows that any set system with n sets and n elements admits a coloring of discrepancy O(n^1/2). Recent exciting work of Bansal, Lovett and Meka shows that such colorings can be found in polynomial time. In…

We prove the following generalised empty pentagon theorem: for every integer $\ell \geq 2$, every sufficiently large set of points in the plane contains $\ell$ collinear points or an empty pentagon. As an application, we settle the next…

Let $P$ be a set of $n$ points in the plane in general position. We show that at least $\lfloor n/3\rfloor$ plane spanning trees can be packed into the complete geometric graph on $P$. This improves the previous best known lower bound…

For the given regular plane polygon and an arbitrary point in the plane of the polygon, the distances from the point to the vertices of the polygon are defined. We proved that there is one more non-congruent regular polygon having the…

We prove that if $N$ points lie in convex position in the plane then they determine $\Omega(N^{5/4})$ distinct angles, provided that the points do not lie on a common circle. This is derived from a more general claim that if $N$ points in…

Let $f(n,k)$ be the minimum number of edges that must be removed from some complete geometric graph $G$ on $n$ points, so that there exists a tree on $k$ vertices that is no longer a planar subgraph of $G$. In this paper we show that…

We propose a geometric structure induced by any given convex polygon $P$, called $Nest(P)$, which is an arrangement of $\Theta(n^2)$ line segments, each of which is parallel to an edge of $P$, where $n$ denotes the number of edges of $P$.…

We consider the problem of computing, given a set S of n points in the plane, which points of S are vertices of the convex hull of S. For certain variations of this problem, different proofs exist that the complexity of this problem in the…

Neumann-Lara and Urrutia showed in 1985 that in any set of n points in the plane in general positionthere is always a pair of points such that any circle through them contains at least (n-2)/60 points. In a series of papers, this result was…

We consider the ring of invariants of n points on the projective line. The space (P^1)^n // PGL_2 is perhaps the first nontrivial example of a Geometry Invariant Theory quotient. The construction depends on the weighting of the n points.…

We consider a minimizing variant of the well-known \emph{No-Three-In-Line Problem}, the \emph{Geometric Dominating Set Problem}: What is the smallest number of points in an $n\times n$~grid such that every grid point lies on a common line…

Let V be a finite set of points in the plane, not contained in a line. Assume |V| = n is an odd number, and |L \cap V| \leq 3 for every line L which is spanned by V. We prove that every simple line L_{a,b} in V creates a simple wedge (i.e.,…

The present work considers the properties of generally convex sets in the $n$-dimensional real Euclidean space $\mathbb{R}^n$, $n>1$, known as weakly $m$-convex, $m=1,2,\ldots,n-1$. An open set of $\mathbb{R}^n$ is called weakly $m$-convex…

Our aim is to prove a Poncelet type theorem for a line configuration on the complex projective. More precisely, we say that a polygon with 2n sides joining 2n vertices A1, A2,..., A2n is well inscribed in a configuration Ln of n lines if…