Oriented Containers - Mixed Questions

We add a little bit to this post.

Sample Question: Which convex region maximizes the difference in orientation between the least area ellipse and least area rectangle that contain it?

Apart from {ellipse, rectangle}, one can look at other pairs of different oriented containers.

Note: An oriented container I didn't talk much about (if at all I mentioned them ever) is the convex lens - formed of two circular arcs which have a common chord. The two chords could be constrained to be of equal radius or left free thereby yielding more questions!