Convex Containers of Triangles

This post is also in continuation of this document written in early 2018:
https://arxiv.org/abs/1802.10447

Given an arbitrary triangle T.

1. How does one find the convex shape C_M of largest area containing T such that T is also the largest area triangle that is contained within C_M?

Guess: for any T, C_M might be some ellipse. However, for a given T, if E is the smallest area ellipse that contains T, T is probably not necessarily the largest triangle that E can contain.
Note: Above question can also be asked with perimeter replacing area.

2. What about the convex shape C_m of smallest area (perimeter) contained within T such that T is also the smallest area (perimeter) triangle that contains C_m?

Guess: again an ellipse.

As usual, one can wonder about 3D. The question might have some connection with Hilbert and Cohn Vossen's statement that any stone will reduce to an ellipsoid on exposure to erosion.
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This question has been asked online: https://mathoverflow.net/questions/345681/on-convex-regions-containing-and-contained-within-a-given-triangle . Further illuminations awaited....
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