Turning a Containers Question Inside Out

This is a Continuation from http://nandacumar.blogspot.com/2021/03/more-on-oriented-containers-and.html

The paper mentioned on that page solves the problem of finding the smallest area isosceles triangle containing a given general triangle.

Basic Question: Given an isosceles triangle T which is the SMALLEST convex region for which T is the smallest area isosceles container?

Now, in this question, one can replace 'isoceles triangle' with rectangle, ellipse, right triangle etc.. and 'smallest area' with 'smallest perimeter' to generate a bunch of questions.

More generally put, the question becomes (https://mathoverflow.net/questions/389865/on-convex-polygons-contained-in-convex-polygons):
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Given a convex n-gon C, find the smallest convex region R such that C is the smallest n-gon that contains it.
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The 2 'smallest's above can independently mean either of 'least area' or 'least perimeter' thus the question is actually 4 questions in one. It seems that for all questions, for any C, R has to touch every side of n along a segment - and thus R should have 2n vertices. Another question which one can ask is if for any C, all or some of the 4 questions have the same R as answer.
Note: If 'smallest' is given other meanings, say 'smallest diameter', there are even more questions in there.