We consider the problem of finding the maximum area parallelogram (MAP) inside a given convex polygon. Our main result is an algorithm for computing the MAP in an n-sided polygon in O(n) time. Achieving this running time requires proving several new structural properties of the MAP. Our algorithm actually computes all the locally maximal area parallelograms (LMAPs). In addition to the algorithm...

F. Chyzak (ed.), INRIA, (2000), pp. ??{??. Enumeration of Geometric Con gurations on a Convex Polygon Marc Noy Universitat Polit ecnica de Catalunya, Barcelona, Espagne December 16, 1999 Summary by Michel Nguy~̂ en-Tĥ e Abstract We survey recent work on the enumeration of non-crossing con gurations on the set of vertices of a convex polygon, such as triangulations, trees, and forests. Exact for...

The celebrated Gauss-Lucas theorem states that all the roots of the derivative of a complex non-constant polynomial p lie in the convex hull of the roots of p, called the Lucas polygon of p. We improve the Gauss-Lucas theorem by proving that all the nontrivial roots of p′ lie in a smaller convex polygon which is obtained by a strict contraction of the Lucas polygon of p. Based on a simple proof...

A new algorithm for the determination of the relative convex hull in the plane of a simple polygon A with respect to another simple polygon B which contains A, is proposed. The relative convex hull is also known as geodesic convex hull, and the problem of its determination in the plane is equivalent to find the shortest curve among all Jordan curves lying in the difference set of B and A and en...

Problem 1: Evaluating Convex Polygons This write-up presents several simple algorithms for determining whether a given set of twodimensional points defines a convex polygon (i.e., a convex hull). In Section 1.1, we introduce the notion of regular polygons and provide examples of both convex and non-convex point sets. Section 1.2 presents an algorithm for ordering a set of points such that a cou...