Topological Art in Simple Galleries

Abstract

Let $P$ be a simple polygon, then the art gallery problem is looking for a minimum set of points (guards) that can see every point in $P$. We say two points $a,b\in P$ can see each other if the line segment $seg(a,b)$ is contained in $P$. We denote by $V(P)$ the family of all minimum guard placements. The Hausdorff distance makes $V(P)$ a metric space and thus a topological space. We show homotopy-universality, that is for every semi-algebraic set $S$ there is a polygon $P$ such that $V(P)$ is homotopy equivalent to $S$. Furthermore, for various concrete topological spaces $T$, we describe instances $I$ of the art gallery problem such that $V(I)$ is homeomorphic to $T$.

32 pages, 36 figures. For associated GeoGebra files, see source files. For associated video, see http://youtube.com/playlist?list=PLh3Niobwkd8pZcSF_Al7e2eeZ-8vqNm-b . Version v2 adds some additional details and references to publications that appeared after v1