Tverberg-type theorems for intersecting by rays

In this paper we consider some results on intersection between rays and a given family of convex, compact sets. These results are similar to the center point theorem, and Tverberg's theorem on partitions of a point set

Coloring geometric hyper-graph defined by an arrangement of half-planes

We prove that any finite set of half-planes can be colored by two colors so that every point of the plane, which belongs to at least three half-planes in the set, is covered by half-planes of both colors. This settles a problem of Keszegh

Ideal boundaries of pseudo-Anosov flows and uniform convergence groups, with connections and applications to large scale geometry

Given a general pseudo-Anosov flow in a three manifold, the orbit space of the lifted flow to the universal cover is homeomorphic to an open disk. We compactify this orbit space with an ideal circle boundary. If there are no perfect fits between stable and unstable leaves and the flow is not topologically conjugate to a suspension Anosov flow, we then show: The ideal circle of the orbit space has a natural quotient space which is a sphere and is a dynamical systems ideal boundary for a compactification of the universal cover of the manifold. The main result is that the fundamental group acts on the flow ideal boundary as a uniform convergence group. Using a theorem of Bowditch, this yields a proof that the fundamental group of the manifold is Gromov hyperbolic and it shows that the action of the fundamental group on the flow ideal boundary is conjugate to the action on the Gromov ideal boundary. This implies that pseudo-Anosov flows without perfect fits are quasigeodesic flows and we show that the stable/unstable foliations of these flows are quasi-isometric. Finally we apply these results to foliations: if a foliation is R-covered or with one sided branching in an atoroidal three manifold then the results above imply that the leaves of the foliation in the universal cover extend continuously to the sphere at infinity.Comment: 69 pages. Major revision, more explanations and simplified some simplified proofs. Detailed explanations of scalloped regions, parabolic points and perfect fit horoballs. 22 figures (3 new figures

Improved bounds for intersecting triangles and halving planes

If a configuration of m triangles in the plane has only n points as vertices, then there must be a set ofmax { [m/(2n - 5)] Ω(m^3 /(n^6 log^2 n))triangles having a common intersection. As a consequence the number of halving planes for a three-dimensional point set is O(n^8/3 log^2/3 n). For all m and n there exist configurations of triangles in which the largest common intersection involvesmax{ [m/(2n - 5)] O(m^2 /n^3)triangles; the upper and lower bounds match for m= O(n^2). The best previous bounds were Ω(m^3 /n^ 6 log^5 n)) for intersecting triangles, and O(n^8/3 log^5/3 n) for halving planes

Retroreflecting curves in nonstandard analysis

We present a direct construction of retroreflecting curves by means of Nonstandard Analysis. We construct non self-intersecting curves which are of class C(1), except for a hyper-finite set of values, such that the probability of a particle being reflected from the curve with the velocity opposite to the velocity of incidence, is infinitely close to 1. The constructed curves are of two kinds: a curve infinitely close to a straight line and a curve infinitely close to the boundary of a bounded convex set. We shall see that the latter curve is a solution of the problem: find the curve of maximum resistance infinitely close to a given curve.CEOCFCTFEDER/POCT

Balanced partitions of 3-colored geometric sets in the plane

Let SS be a finite set of geometric objects partitioned into classes or colors . A subset S'¿SS'¿S is said to be balanced if S'S' contains the same amount of elements of SS from each of the colors. We study several problems on partitioning 33-colored sets of points and lines in the plane into two balanced subsets: (a) We prove that for every 3-colored arrangement of lines there exists a segment that intersects exactly one line of each color, and that when there are 2m2m lines of each color, there is a segment intercepting mm lines of each color. (b) Given nn red points, nn blue points and nn green points on any closed Jordan curve ¿¿, we show that for every integer kk with 0=k=n0=k=n there is a pair of disjoint intervals on ¿¿ whose union contains exactly kk points of each color. (c) Given a set SS of nn red points, nn blue points and nn green points in the integer lattice satisfying certain constraints, there exist two rays with common apex, one vertical and one horizontal, whose union splits the plane into two regions, each one containing a balanced subset of SS.Peer ReviewedPostprint (published version