O. Aichholzer, M. Cetina, R. Fabila-Monroy, J. Leaños, G. Salazar,
and J. Urrutia
Let
be a simple polygon on the plane. Two vertices of
are visible if
the open line segment joining them is contained in the interior of
. 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.