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