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Reference: https://en.wikipedia.org/wiki/N-ellipse

Question: How does one find and characterize the smallest 3-ellipses (n-ellipses with n =3) that contain a given triangle? 'Smallest' can mean 'least area' or 'least perimeter' or... and may have different answers. Are 3-ellipses for which the 3 vertices of the triangle are themselves the foci good candidates?

And what about the largest 3-ellipses inscribed in a given triangle?

Note: Versions of these questions for n>3 and 3-d can also be considered. Maybe one can prove (say): smallest n+1-ellipse containing any triangle is smaller than the smallest n-ellipse containing the triangle.

Variants (September 10th, 2021): Instead of 3-ellipses - and multifocal ellipses - one can ask the above questions with convex Cartesian Ovals (with 2 or more foci). Reference: https://en.wikipedia.org/wiki/Cartesian_oval

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My answer here may help, especially the citation to

Nie, Jiawang, Pablo A. Parrilo, and Bernd Sturmfels. "Semidefinite representation of the $k$-ellipse." In Algorithms in Algebraic Geometry, pp. 117-132. Springer, New York, NY, 2008.

   

The polynomial for a $3$-ellipse has degree-$8$. "We express this polynomial equation as the determinant of a symmetric matrix of linear polynomials."

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    $\begingroup$ The Wikipedia image showing the nested $3$-ellipses for increasing distances suggests it would not be difficult to binary search for the (approximate) distance such that the $3$-ellipse includes two foci and passes through the third. This would be the smallest $3$-ellipse. $\endgroup$ Sep 7 '21 at 23:52
  • $\begingroup$ Thank you. Another possibility is probably to consider only 3-ellipses which pass thru all vertices of the triangle - but then, it seems at most only one of the vertices can be a focus and finding the position of the other foci would become another problem! $\endgroup$ Sep 8 '21 at 5:24

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