Given an arbitrary triangle T.
- How does one find the convex region 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 be asked with 'perimeter' replacing 'area'. 'Mixed questions' can also be posed - ".....C_M of largest area containing T such that T is the largest perimeter triangle....."
- What about the convex shape C_m of smallest area contained within T such that T is also the smallest area triangle that contains C_m?
Note: This question too can have a 'perimeter version' and 'mixed versions'.
Guess: again an ellipse.
Remarks: These questions can be asked with 'rectangle' or 'ellipse' replacing 'triangle'. And raised to 3D, the question might have some connection with Hilbert and Cohn Vossen's statement that any stone will reduce to an ellipsoid on prolonged exposure to erosion.
Additional note(added on 22 November 2019): This is on the question: "Given a rectangle R, find that convex region C containing it with maximum area such that R is the rectangle of maximum area contained within C. " Our experiments strongly indicate that for any R, C is an ellipse. However, with 'perimeter' replacing 'area' in this question, the indications are that C may not be an ellipse.