Let P be a set of points in general position in the plane. Join all pairs of points in P with straight line segments. The number of segment-crossings in such a drawing, denoted by cr(P ), is the rectilinear crossing number of P . A halving line of P is a line passing through two points of P that divides the rest of the points of P in (almost) half. The number of halving lines of P is denoted by h(P ). Similarly, a k-edge, 0 ≤ k ≤ n/2 − 1, is a line passing through two points of P and leaving exactly k points of P on one side. The number of ≤ k-edges of P is denoted by E ≤k (P ). Let cr(n), h(n), and E ≤k (n) denote the minimum of cr(P ), the maximum of h(P ), and the minimum of E ≤k (P ), respectively, over all sets P of n points in general position in the plane. We show that the previously best known lower bound on E ≤k (n) is tight for k < (4n − 2)/9 and improve it for all k ≥ (4n − 2)/9 . This in turn improves the lower bound on cr(n) from 0.37968 n 4 + Θ(n 3 ) to 277 729 n 4 + Θ(n 3 ) ≥ 0.37997 n 4 + Θ(n 3 ). We also give the exact values of cr(n) and h(n) for all n ≤ 27. B.M. Ábrego ( ) · S. Fernández-Merchant Discrete Comput Geom (2012) 48:192-215 193Exact values were known only for n ≤ 18 and odd n ≤ 21 for the crossing number, and for n ≤ 14 and odd n ≤ 21 for halving lines.

Even the most supercial glance at the vast majority of crossing-minimal geometric drawings of K n reveals two hard-to-miss features. First, all such drawings appear to be 3-fold symmetric (or simply 3-symmetric). And second, they all are 3-decomposable, that is, there is a triangle T enclosing the drawing, and a balanced partition A, B, C of the underlying set of points P , such that the orthogonal projections of P onto the sides of T show A between B and C on one side, B between A and C on another side, and C between A and B on the third side. In fact, we conjecture that all optimal drawings are 3-decomposable, and that there are 3-symmetric optimal constructions for all n multiple of 3. In this paper, we show that any 3-decomposable geometric drawing of K n has at least 0.380029 n 4 + Θ(n 3) crossings. On the other hand, we produce 3-symmetric and 3-decomposable drawings that improve the general upper bound for the rectilinear crossing number of K n to 0.380488 n 4 + Θ(n 3). We also give explicit 3-symmetric and 3-decomposable constructions for n < 100 that are at least as good as those previously known.

Abstract. Let P be a simple polygon on the plane. Two vertices of P are visible if the open line segment joining them is contained in the interior of P . In this paper we study the following questions posed in [7,8]: (1) Is it true that every non-convex simple polygon has a vertex that can be continuously moved such that during the process no vertex-vertex visibility is lost and some vertex-vertex visibility is gained? (2) Can every simple polygon be convexified by continuously moving only one vertex at a time without losing any internal vertex-vertex visibility during the process? We provide a counterexample to (1). We note that our counterexample uses a monotone polygon. We also show that question (2) has a positive answer for monotone polygons.

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Abstract. Devadoss asked: (1) can every polygon be convexified so that no internal visibility (between vertices) is lost in the process? Moreover, (2) does such a convexification exist, in which exactly one vertex is moved at a time (that is, using single-vertex moves)? We prove the redundancy of the "singlevertex moves" condition: an affirmative answer to (1) implies an affirmative answer to (2). Since Aichholzer et al. recently proved (1), this settles (2).