In the post 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):
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?
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 On convex regions containing (and contained within) a given triangle.
Related discussions: On Convex Regions Contained in Convex Polygons, On convex polygons contained in convex polygons, Smallest 3-ellipses that contain triangles
