In the post https://mathoverflow.net/questions/345681/on-convex-regions-containing-and-contained-within-a-given-triangle , it was noted:
-   for a general triangle T, the convex region C_M of largest area containing  T such that T is  the largest area triangle that is contained within C_M is the Steiner circumellipse of T.

**New Questions** (generalizing from triangle): 
1. Given a general convex quadrilateral Q, which is the convex region C_M of largest area containing Q such that Q  is also the largest area convex quadrilateral contained within C_M? Is it a 3-ellipse? If so, does this generalize to convex n-gons and (n-1)-ellipses?

2. What about the convex region C_m of least area contained in Q such that Q is also the smallest area quadrilateral containing C_m? Is it too a 3-ellipse?

Note: I do not know the answers to questions that can be generated with perimeter replacing area in any of the above questions including those asked in https://mathoverflow.net/questions/345681/on-convex-regions-containing-and-contained-within-a-given-triangle.  

Related discussions: 
https://mathoverflow.net/questions/393174/on-convex-regions-contained-in-convex-polygons, 
https://mathoverflow.net/questions/389865/on-convex-polygons-contained-in-convex-polygons, https://mathoverflow.net/questions/403370/smallest-3-ellipses-that-contain-triangles