A new upper bound for the VC-dimension of visibility regions
Alexander Gilbers, Rolf Klein · 2011
In this paper we are proving the following fact. Let P be an arbitrary simple polygon, and let S be an arbitrary set of 15 points inside P. Then there exists a subset T of S that is not "visually discernible", that is, T ≠ vis(v) ∩ S holds for the visibility regions vis(v) of all points v in P. In other words, the VC-dimension $d$ of visibility regions in a simple polygon cannot exceed 14. Since Valtr [v-ggwps-98] proved in 1998 that d ∈ [6,23] holds, no progress has been made on this bound. Our reduction immediately implies a smaller upper bound to the number of guards needed to cover P by ε-net theorems.