The one-round Voronoi game replayed

We resolve a number of open problems raised by that paper. In particular, we give a precise characterization of the outcome of the game for optimal play: we show that Barney has a winning strategy for n⩾3 and ρ>2/n, and for n=2 and ρ>3/2. Wilma wins in all remaining cases, i.e., for n⩾3 and ρ⩽2/n, for n=2 and ρ⩽3/2, and for n=1. We also discuss complexity aspects of the game on more general boards, by proving that for a polygon with holes, it is NP-hard to maximize the area Barney can win against a given set of points by Wilma.