On Grids in Point-Line Arrangements in the Plane

The famous Szemer\'{e}di-Trotter theorem states that any arrangement of $n$ points and $n$ lines in the plane determines $O(n^{4/3})$ incidences, and this bound is tight. In this paper, we prove the following Tur\'an-type result for point-line incidence. Let $\mathcal{L}_1$ and $\mathcal{L}_2$ be two sets of $t$ lines in the plane and let $P=\{\ell_1 \cap \ell_2 : \ell_1 \in \mathcal{L}_1, \ell_2 \in \mathcal{L}_2\}$ be the set of intersection points between $\mathcal{L}_1$ and $\mathcal{L}_2$. We say that $(P, \mathcal{L}_1 \cup \mathcal{L}_2)$ forms a \emph{natural $t\times t$ grid} if $|P| =t^2$, and $conv(P)$ does not contain the intersection point of some two lines in $\mathcal{L}_i,$ for $i = 1,2.$ For fixed $t > 1$, we show that any arrangement of $n$ points and $n$ lines in the plane that does not contain a natural $t\times t$ grid determines $O(n^{\frac{4}{3}- \varepsilon})$ incidences, where $\varepsilon = \varepsilon(t)$. We also provide a construction of $n$ points and $n$ lines in the plane that does not contain a natural $2 \times 2$ grid and determines at least $\Omega({n^{1+\frac{1}{14}}})$ incidences.Comment: 13 pages, 5 figure

Dense point sets have sparse Delaunay triangulations

The spread of a finite set of points is the ratio between the longest and shortest pairwise distances. We prove that the Delaunay triangulation of any set of n points in R^3 with spread D has complexity O(D^3). This bound is tight in the worst case for all D = O(sqrt{n}). In particular, the Delaunay triangulation of any dense point set has linear complexity. We also generalize this upper bound to regular triangulations of k-ply systems of balls, unions of several dense point sets, and uniform samples of smooth surfaces. On the other hand, for any n and D=O(n), we construct a regular triangulation of complexity Omega(nD) whose n vertices have spread D.Comment: 31 pages, 11 figures. Full version of SODA 2002 paper. Also available at http://www.cs.uiuc.edu/~jeffe/pubs/screw.htm

An Improved Bound for Weak Epsilon-Nets in the Plane

We show that for any finite set $P$ of points in the plane and $\epsilon>0$ there exist $\displaystyle O\left(\frac{1}{\epsilon^{3/2+\gamma}}\right)$ points in ${\mathbb{R}}^2$, for arbitrary small $\gamma>0$, that pierce every convex set $K$ with $|K\cap P|\geq \epsilon |P|$. This is the first improvement of the bound of $\displaystyle O\left(\frac{1}{\epsilon^2}\right)$ that was obtained in 1992 by Alon, B\'{a}r\'{a}ny, F\"{u}redi and Kleitman for general point sets in the plane.Comment: A preliminary version to appear in the proceedings of FOCS 201

Discrete Geometry

The workshop on Discrete Geometry was attended by 53 participants, many of them young researchers. In 13 survey talks an overview of recent developments in Discrete Geometry was given. These talks were supplemented by 16 shorter talks in the afternoon, an open problem session and two special sessions. Mathematics Subject Classification (2000): 52Cxx. Abstract regular polytopes: recent developments. (Peter McMullen) Counting crossing-free configurations in the plane. (Micha Sharir) Geometry in additive combinatorics. (József Solymosi) Rigid components: geometric problems, combinatorial solutions. (Ileana Streinu) • Forbidden patterns. (János Pach) • Projected polytopes, Gale diagrams, and polyhedral surfaces. (Günter M. Ziegler) • What is known about unit cubes? (Chuanming Zong) There were 16 shorter talks in the afternoon, an open problem session chaired by Jesús De Loera, and two special sessions: on geometric transversal theory (organized by Eli Goodman) and on a new release of the geometric software Cinderella (Jürgen Richter-Gebert). On the one hand, the contributions witnessed the progress the field provided in recent years, on the other hand, they also showed how many basic (and seemingly simple) questions are still far from being resolved. The program left enough time to use the stimulating atmosphere of the Oberwolfach facilities for fruitful interaction between the participants

A Szemeredi-Trotter type theorem in R4\mathbb{R}^4

We show that $m$ points and $n$ two-dimensional algebraic surfaces in $\mathbb{R}^4$ can have at most $O(m^{\frac{k}{2k-1}}n^{\frac{2k-2}{2k-1}}+m+n)$ incidences, provided that the algebraic surfaces behave like pseudoflats with $k$ degrees of freedom, and that $m\leq n^{\frac{2k+2}{3k}}$. As a special case, we obtain a Szemer\'edi-Trotter type theorem for 2--planes in $\mathbb{R}^4$, provided $m\leq n$ and the planes intersect transversely. As a further special case, we obtain a Szemer\'edi-Trotter type theorem for complex lines in $\mathbb{C}^2$ with no restrictions on $m$ and $n$ (this theorem was originally proved by T\'oth using a different method). As a third special case, we obtain a Szemer\'edi-Trotter type theorem for complex unit circles in $\mathbb{C}^2$. We obtain our results by combining several tools, including a two-level analogue of the discrete polynomial partitioning theorem and the crossing lemma.Comment: 50 pages. V3: final version. To appear in Discrete and Computational Geometr

The visible perimeter of an arrangement of disks

Given a collection of n opaque unit disks in the plane, we want to find a stacking order for them that maximizes their visible perimeter---the total length of all pieces of their boundaries visible from above. We prove that if the centers of the disks form a dense point set, i.e., the ratio of their maximum to their minimum distance is O(n^1/2), then there is a stacking order for which the visible perimeter is Omega(n^2/3). We also show that this bound cannot be improved in the case of a sufficiently small n^1/2 by n^1/2 uniform grid. On the other hand, if the set of centers is dense and the maximum distance between them is small, then the visible perimeter is O(n^3/4) with respect to any stacking order. This latter bound cannot be improved either. Finally, we address the case where no more than c disks can have a point in common. These results partially answer some questions of Cabello, Haverkort, van Kreveld, and Speckmann.Comment: 12 pages, 5 figure

Debris Disks: Seeing Dust, Thinking of Planetesimals and Planets

Debris disks are optically thin, almost gas-free dusty disks observed around a significant fraction of main-sequence stars older than about 10 Myr. Since the circumstellar dust is short-lived, the very existence of these disks is considered as evidence that dust-producing planetesimals are still present in mature systems, in which planets have formed - or failed to form - a long time ago. It is inferred that these planetesimals orbit their host stars at asteroid to Kuiper-belt distances and continually supply fresh dust through mutual collisions. This review outlines observational techniques and results on debris disks, summarizes their essential physics and theoretical models, and then places them into the general context of planetary systems, uncovering interrelations between the disks, dust parent bodies, and planets. It is shown that debris disks can serve as tracers of planetesimals and planets and shed light on the planetesimal and planet formation processes that operated in these systems in the past.Comment: Review paper, accepted for publication in "Research in Astronomy and Astrophysics