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 $$a,b\in P$$ a , b ∈ P can see each other if the line segment $${\text {seg}} (a,b)$$ 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 .