Greedy Routing is a class of routing algorithms in which the packets are forwarded in a manner that reduces the distance to the destination at every step. In an attempt to provide theoretical guarantees for a class of greedy routing algorithms, Papadimitriou and Ratajczak (Theor. Comput. Sci. 344(1):3-14, 2005) came up with the following conjecture:Any 3-connected planar graph can be drawn in the plane such that for every pair of vertices s and t a distance decreasing path can be found.We settle this conjecture in the affirmative for the case of triangulations. A partitioning of the edges of a triangulation G into 3 trees, called the realizer of G, was first developed by Schnyder who also gave a drawing algorithm based on this. We generalize Schnyder's algorithm to obtain a whole class of drawings of any given triangulation G. We show, using the Knaster-Kuratowski-Mazurkiewicz Theorem, that some drawing of G belonging to this class is greedy.

We extend to topological affine planes the standard theorems of convexity, among them the separation theorem, the anti-exchange theorem, Radon's, Helly's, Carathéodory's, and Kirchberger's theorems, and the Minkowski theorem on extreme points. In a few cases the proofs are obtained by adapting proofs of the original results in the Euclidean plane; in others it is necessary to devise new proofs that are valid in the more general setting considered here. Basic DefinitionsThe notion of a topological affine plane (TAP) A is most simply defined by means of one of its standard models (see, e.g., [7]): Consider a circle C ∞ in the Euclidean plane; its interior, int C ∞ , constitutes the set |A| of points of A. For each pair of antipodal points on C ∞ , the interior of a simple Jordan arc joining the points (and not meeting C ∞ anywhere else except at its endpoints) is called a pseudoline. Suppose we are given, for each pair x, y of points of |A|, a unique pseudoline ← →x y containing x and y and depending * Jacob E.

Abstract. We prove that the number of distinct weaving patterns produced by n semi-algebraic curves in R 3 defined coordinate-wise by polynomials of degrees bounded by some constant d, is bounded by 2 O(n log n) , where the implied constant in the exponent depends on d. This generalizes a similar bound obtained by Pach, Pollack and Welzl [3] for the case when d = 1.