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):

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 On convex regions containing (and contained within) a given triangle.

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

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):

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 On convex regions containing (and contained within) a given triangle.

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

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):

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 On convex regions containing (and contained within) a given triangle.

Related discussions: https://mathoverflow.net/questions/393174/on-convex-regions-contained-in-convex-polygons, On convex polygons contained in convex polygons

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):

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 On convex regions containing (and contained within) a given triangle.

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