Given a simple polygon in the plane, a deflation is defined as the inverse of a flip in the Erdős-Nagy sense. In 1993 Bernd Wegner conjectured that every simple polygon admits only a finite number of deflations. In this note we describe a counter-example to this conjecture by exhibiting a family

Given simple polygons P and Q, their separation, denoted by (P; Q), is de ned to be the minimum distance between their boundaries. We present a parallel algorithm for finding a closest pair among all pairs (p; q), p 2 P and q 2 Q. The algorithm runs in O(log n) time using O(n) processors on a CREW

It is well known that, given two simple n-sided polygons, it may not be possible to triangulate the two polygons in a compatible fashion, if one's choice of triangulation vertices is restricted to polygon corners. Is it always possible to produce compatible triangulations if additional

We discuss problems of optimizing the area of a simple polygon for a given set of vertices P and show that these problems are very closely related to problems of optimizing the number of points from a set Q in a simple polygon with vertex set P . We prove that it is NP-complete to find a minimum

We discuss the problem of nding a simple polygonalization for a given set of vertices P that has optimal area. We show that these problems are very closely related to problems of optimizing the number of points from a set Q in a simple polygon with vertex set P and prove that it is NP

We present a new method of decomposing a simple polygon which allows the preprocessing of the polygon to efficiently answer queries of various forms. Using ) space, we can, given a query point q inside or outside the polygon, recover the number of vertices visible from q in O(log n) time. Also

We prove a conjecture of Aanjaneya, Bishnu, and Pal that the maximum number of diffuse reflections needed for a point light source to illuminate the interior of a simple polygon with n walls is ⌊n/2 ⌋ − 1. Light reflecting diffusely leaves a surface in all directions, rather than at an identical

We prove a conjecture of Aanjaneya, Bishnu, and Pal that the maximum number of diffuse reflections needed for a point light source to illuminate the interior of a simple polygon with n walls is bn/2c − 1. Light reflecting diffusely leaves a surface in all directions, rather than at an identical

This paper considers the topic of efficiently triangulating a simple polygon with emphasis on practical and easy-to-implement algorithms. It also describes a new adaptive algorithm for triangulating a simple n-sided polygon. The algorithm runs in time O(n(1+t 0 )), with t 0 < n

that in Euclidean spaces, the Pareto envelopes and the convex hulls coincide. Chalmet et al. (1981) characterized the Pareto envelopes in the rectilinear plane (R2, d1) and constructed them in O(n log n) time. In this note, we investigate the Pareto envelopes of point-sets in simple polygons P endowed with geodesic