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