Polygons Needing Many Flipturns

A flipturn on a polygon consists of reversing the order of edges inside a pocket of the polygon, without changing lengths or slopes. Any polygon with n edges must be convexified after at most (n− 1)! flipturns. A recent paper showed that in fact it will be convex after at most n(n−3)/2 flipturns. We give here lower bounds. We construct a polygon such that if pockets are chosen in a bad way, at least (n − 2)2/4 flipturns are needed to convexify the polygon. In another construction, (n − 1)2/8 flipturns are needed, regardless of the order in which pockets are chosen. All our bounds are adaptive to a pre-specified number of distinct slopes of the edges.