We consider planar geometric models given by an explicit boundary of 0 en) algebraic curve segments of maximum degree d. We present an 0 en"dO(l) time algorithm to compute its convex hull and an 0 (nlog logn +K) . dO(l) time algorithms to compute various decompositions of an object, where K is the characteristic number of this object. Both operations, besides being solutions to interesting comp...

Let S be a set of n points moving on the real line. The kinetic sorting problem is to maintain a data structure on the set S that makes it possible to quickly generate a sorted list of the points in S, at any given time. We prove tight lower bounds for this problem, which show the following: with a subquadratic maintenance cost one cannot obtain any significant speed-up on the time needed to ge...

Lifts/extended formulations/cone representations of convex sets currently form an active area of research in optimization, computer science, real algebraic geometry and convex geometry. We invite high quality papers on all optimization related aspects of this topic for a special issue of Mathematical Programming, Series B. All submitted papers will be refereed according to the standards of Math...

A convex hole (or empty convex polygon) of a point set P in the plane is a convex polygon with vertices in P , containing no points of P in its interior. Let R be a bounded convex region in the plane. We show that the expected number of vertices of the largest convex hole of a set of n random points chosen independently and uniformly over R is Θ(logn/(log log n)), regardless of the shape of R.

A convex subset of vertices of a tournament T is such that any vertex not in the subset either dominates or is dominated by all of the vertices in the convex subset. Given a large tournament T , we explore the structure of T by looking for its transitive convex subsets. In a majority voting tournament some isolated inconsistencies may appear in such subsets. We propose a way to reveal such “nea...

Let R and B be disjoint point sets such that R ∪ B is in general position. We say that B is enclosed by R if there is a simple polygon P with vertex set R such that all the elements in B belong to the interior of P . In this paper we prove that if the vertices of the convex hull of R ∪B belong to B, and |R| ≤ |Conv(R)| − 1 then B encloses R. The bound is tight. This improves on results of a pre...

3 Let S be a set of n points in the plane and let R be a set of n pairwise non-crossing rays, 4 each with an apex at a different point of S. Two sets of non-crossing rays R1 and R2 are 5 considered to be different if the cyclic permutations they induce at infinity are different. 6 In this paper, we study the number r(S) of different configurations of non-crossing rays 7 that can be obtained fro...

Data imprecision constitutes an important gap between theory and practice in computational geometry. A lot of research about imprecision in computational geometry is directed at computing the convex hull of imprecise points rather than imprecise line segment intersection. In this paper we introduce an algorithm to construct the convex hull for a set of imprecise line segment intersection in time.