Given an arbitrary triangle T.

  1. 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....."

  1. 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.

  • 2
    $\begingroup$ 1) Yes, this is the ellipse which is a disc if $T$ is equilateral (and the corresponding affine image in arbitrary case). The claim is equivalent to the following: if $\lambda=\frac{3\sqrt{3}}{4\pi}$ is the ratio of areas of the equilateral triangle inscribed to a circle and the interior of the circle, then any convex set of unit area contains a triangle of area not less than $\lambda$. This is well-known and may be proved by Steiner symmetrization. $\endgroup$ – Fedor Petrov Nov 10 at 10:43
  • $\begingroup$ 2) Analogously, the claim is that any convex set of unit area is contained in a triangle of area $3\sqrt{3}/\pi$ (equality for a disc). This should be also known, but unfortunately not for me. $\endgroup$ – Fedor Petrov Nov 10 at 10:44
  • $\begingroup$ Could you clarify if the answer to question 1 is the Steiner circumellipse? Is the following correct: "Among all triangles inscribed inside any given ellipse E, those triangles for which E is the Steiner circumellipse are the ones with largest area"? $\endgroup$ – Nandakumar R Nov 10 at 12:57
  • $\begingroup$ Also hope to hear on the 'perimeter' versions of the question. $\endgroup$ – Nandakumar R Nov 10 at 12:58
  • $\begingroup$ The answer to question 1 is the circumscribed ellipse which centre is the barycentre of $T$. If it is called Steiner circumellipse then yes. $\endgroup$ – Fedor Petrov Nov 10 at 16:40

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