Convexifying monotone polygons while maintaining internal visibility

O. Aichholzer, M. Cetina, R. Fabila-Monroy, J. Leaños, G. Salazar, and J. Urrutia

Abstract:

Let $P$ be a simple polygon on the plane. Two vertices of $P$ are visible if the open line segment joining them is contained in the interior of $P$. In this paper we study the following questions posed by Devadoss: (1) Is it true that every non-convex simple polygon has a vertex that can be continuously moved such that during the process no vertex-vertex visibility is lost and some vertex-vertex visibility is gained? (2) Can every simple polygon be convexified by continuously moving only one vertex at a time without losing any internal vertex-vertex visibility during the process? We provide a counterexample to (1). We note that our counterexample uses a monotone polygon. We also show that question (2) has a positive answer for monotone polygons.

Reference: O. Aichholzer, M. Cetina, R. Fabila-Monroy, J. Leaños, G. Salazar, and J. Urrutia. Convexifying monotone polygons while maintaining internal visibility. In A. Marquez, P. Ramos, and J. Urrutia, editors, Special issue: XIV Encuentros de Geometría Computacional ECG2011, volume 7579 of Lecture Notes in Computer Science (LNCS), pages 98-108. Springer, 2012.