Before revision:  (} 2) how to such that \Theta}\ge compute and maintain the rotated hull \mathcal{OH_{\theta}(P ) }  }} for \theta\in [0,2\pi) in O(kn\log n) time and O(kn) space, and (3) how to compute in \Theta(n{\Theta}} \log n) time and O( n) space a value of \theta for which the rectilinear convex hull, \mathcal{RH_{\theta}(P ), has minimum area, thus improving the previously best O(n^2) algorithm presented by Bae et al.in 2009.
After revision:  (} 2) how to such that \Theta}\ge\frac{\pi compute and maintain the rotated hull \mathcal{OH_{\theta}(P ) } }} for \theta\in [0,2\pi) in O(kn\log n) time and O(kn) space, and (3) how to compute in \Theta(n{\Theta}} \log n) time and O( n) space a value of \theta for which the rectilinear convex hull, \mathcal{RH_{\theta}(P ), has minimum area, thus improving the previously best O(n^2) algorithm presented by Bae et al.in 2009.