We consider a competitive facility location problem with two players. Players alternate placing points, one at a time, into the playing arena, until each of them has placed n points. The arena is then subdivided according to the nearest-neighbor rule, and the player whose points control the larger area wins. We present a winning strategy for the second player, where the arena is a circle or a line segment. We permit variations where players can play more than one point at a time, and show that the ÿrst player can ensure that the second player wins by an arbitrarily small margin.

Given two points in a simple polygon P of n vertices, its geodesic distance is the length of the shortest path that connects them among all paths that stay within P . The geodesic center of P is the unique point in P that minimizes the largest geodesic distance to all other points of P . In 1989, Pollack, Sharir and Rote [Disc. & Comput. Geom. 89] showed an O(n log n)-time algorithm that computes the geodesic center of P . Since then, a longstanding question has been whether this running time can be improved (explicitly posed by Mitchell [Handbook of Computational Geometry, 2000]). In this paper we affirmatively answer this question and present a linear time algorithm to solve this problem.

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In casting, liquid is poured into a cast that has a cavity with the shape of the object to be manufactured. The liquid then hardens, after which the cast is removed. We consider the case where the cast consists of two parts and address the following problems: 1 Given a cast for an object and a directiond, can the cast be partitioned into two parts such that the parts can be removed in directionsd and ,d, respectively, without colliding with the object or the other cast part? 2 How can one nd a directiond such that the above cast partitioning can be done? We give necessary and su cient conditions for both problems, as well as algorithms to decide them for polyhedral objects. We also give some evidence that the case where the cast parts need not be removed in opposite directions is considerably harder.

We study the problems of computing two non-convex enclosing shapes with the minimum area; the L-shape and the rectilinear convex hull. Given a set of n points in the plane, we find an L-shape enclosing the points or a rectilinear convex hull of the point set with minimum area over all orientations. We show that the minimum enclosing shapes for fixed orientations change combinatorially at most O (n) times while rotating the coordinate system. Based on this, we propose efficient algorithms that compute both shapes with the minimum area over all orientations. The algorithms provide an efficient way of maintaining the set of extremal points, or the staircase, while rotating the coordinate system, and compute both minimum enclosing shapes in O (n 2 ) time and O (n) space. We also show that the time complexity of maintaining the staircase can be improved if we use more space.

Given a set of sites in a simple polygon, a geodesic Voronoi diagram of the sites partitions the polygon into regions based on distances to sites under the geodesic metric. We present algorithms for computing the geodesic nearest-point, higher-order and farthest-point Voronoi diagrams of m point sites in a simple n-gon, which improve the best known ones for m ≤ n/ polylog n. Moreover, the algorithms for the geodesic nearest-point and farthest-point Voronoi diagrams are optimal for m ≤ n/ polylog n. This partially answers a question posed by Mitchell in the Handbook of Computational Geometry.

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Given a planar convex set C, we give sublinear approximation algorithms to determine approximations of the largest axially symmetric convex set S contained in C, and the smallest such set S that contains C. More precisely, for any ε > 0, we find an axially symmetric convex polygon Q ⊂ C with area |Q| > (1 − ε)|S| and we find an axially symmetric convex polygon Q containing C with area |Q | < (1 + ε)|S |. We assume that C is given in a data structure that allows to answer the following two types of query in time T C : given a direction u, find an extreme point of C in direction u, and given a line , find C ∩. For instance, if C is a convex n-gon and its vertices are given in a sorted array, then T C = O(log n). Then we can find Q and Q in time O(ε −1/2 T C + ε −3/2). Using these techniques, we can also find approximations to the perimeter, area, diameter, width, smallest enclosing rectangle and smallest enclosing circle of C in time O(ε −1/2 T C).