Related papers: On separability in discrete geometry

We show an equivalence between a conjecture of Bisztriczky and Fejes T{\'o}th about arrangements of planar convex bodies and a conjecture of Goodman and Pollack about point sets in topological affine planes. As a corollary of this…

In this paper we consider the following topics related to results of L\'aszl\'o Fejes T\'oth: (1) The Tammes problem and Fejes T\'oth's bound on circle packings; (2) Fejes T\'oth's problem on maximizing the minimum distance between…

A problem posed by Erd\H{o}s in 1945 initiated the study of non-separable arrangements of convex bodies. A finite collection of convex bodies in Euclidean $d$-space is called a non-separable family (or NS-family) if every hyperplane…

A packing of translates of a convex domain in the Euclidean plane is said to be totally separable if any two packing elements can be separated by a line disjoint from the interior of every packing element. This notion was introduced by G.…

We give new bounds on the Erdos-Szekeres theorems for convex bodies of Bisztriczky and Fejes Toth and of Pach and Toth. We derive them from a combinatorial characterization of convex position of a family of planar convex bodies. This…

At a first glance, the problem of illuminating the boundary of a convex body by external light sources and the problem of covering a convex body by its smaller positive homothetic copies appear to be quite different. They are in fact two…

In 1959 Fejes T\'oth posed a conjecture that the sum of pairwise non-obtuse angles between $N$ unit vectors in $\mathbb S^d$ is maximized by periodically repeated elements of the standard orthonormal basis. We obtain new improved upper…

In discrete geometry, the contact number of a given finite number of non-overlapping spheres was introduced as a generalization of Newton's kissing number. This notion has not only led to interesting mathematics, but has also found…

A discretisation scheme that preserves topological features of a physical problem is extended so that differential geometric structures can be approximated in a consistent way thus giving access to the study of physical systems which are…

A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete…

In geometry, there are several challenging problems studying numbers associated to convex bodies. For example, the packing density problem, the kissing number problem, the covering density problem, the packing-covering constant problem,…

Erd\"os proved in 1946 that if a set $E\subset\mathbb{R}^n$ is closed and non-empty, then the set, called ambiguous locus or medial axis, of points in $\mathbb{R}^n$ with the property that the nearest point in $E$ is not unique, can be…

Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a `reduction' of the `universal differential algebra' and this allows a systematic…

The purpose of this paper is to study the reflections of a convex body. In particular, we are interested in orthogonal reflections of its sections that can be extended to reflections of the whole body. For this reason, we need to study the…

This text has three parts. The first one is largely autobiographical, hence my use of the first person. There I recall how Gerard Cohen influenced important parts of my research. The second is of a more classic mathematical nature. I…

We study the order dimension of the lattice of closed sets for a convex geometry. Further, we prove the existence of large convex geometries realized by planar point sets that have very low order dimension. We show that the planar point set…

The first part of this article is a short and selective survey of developments in differential and algebraic geometry from the 1980's involving enumerative questions and nonlinear elliptic partial differential equations. In the second part…

Let $K_0$ be a compact convex subset of the plane $\mathbb R^2$, and assume that whenever $K_1\subseteq \mathbb R^2$ is congruent to $K_0$, then $K_0$ and $K_1$ are not crossing in a natural sense due to L. Fejes-T\'oth. A theorem of L.…

In 1997, Erd\H{o}s asked whether for arbitrarily large $n$ there exists a set of $n$ points in $\mathbb{R}^2$ that determines $O(\frac{n}{\sqrt{\log n}})$ distinct distances while satisfying the local constraint that every 4-point subset…

The Erd\H{o}s distinct distance problem is a ubiquitous problem in discrete geometry. Somewhat less well known is Erd\H{o}s' distinct angle problem, the problem of finding the minimum number of distinct angles between $n$ non-collinear…