The Maximum Number of Ways to Stab N Convex Non Intersecting Objects in the Plan

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The Maximum Number of Ways to Stab N Convex Non Intersecting Objects in the Plan
Herbert Edelsbrunner
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'Courant Institute of Mathematical Sciences, New York University, New York, N'Y 10012, U. SwA. ., and School of . Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel. 1. Introduction Let S be a set of n convex, closed, bounded, and pairwise non-intersecting objects in the Euclidean plane. We label the objects with the integers from 1 through n. A directed or undirected line that intersects all objects is called a transversal of S. Since no two objects intersect each other, a transversa...l intersects 5 in a well- defined order. In the case of an undirected line, such an order can be described by a pair of permutations, one being the reverse of the other. Such a pair is called a geometric permutation of S. For convenience, we will represent a geometric per- mutation by any one of its two permutations. For example, let S consist of two equally large disks, labeled n— 1 and n, and of n— 2 horizontal line segments, labeled from 1 through n— 2. The two disks are placed sufficiently close to each other such that their centers lie on a common hor- izontal line h.

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