n-ellipse questions

The following are queries put up last week at mathorverflow:
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Qn. Given any 3-ellipse. Is the largest circle contained in this 3-ellipse unique (ie there is only one largest inscribed circle)? If so, is it generally true that for any m < n, the largest area (or perimeter) m-ellipse contained in any given n-ellipse is unique?

Note 1: The largest area and largest perimeter contained m-ellipses in a given n-ellipse may not always be the same but I have no convincing example.

Note 2: Analogously, one can ask whether least area and least perimeter m-ellipses that contain a given n-ellipse can be different.
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This was a well-received post at mathoverflow. One can ask some further similar questions by replacing circles/ellipses with 3-ellipses, n-ellipses and curves of constant width.

Triangle Centers - a consolidated record

Pulling together some strands of thought from this post and this and the previous post and applying them specifically to triangles:

The centers contemplated given a general triangle T:

1. For each point P in the interior of T, define the ratio: length of longest chord thru a point / length of shortest chord thru that point. Find where it is a minimum. (Instead of ratio one can minimize the difference between these two quantities and find another center).

2. For each point P in the interior of T, consider the ratio: distance of farther point on the boundary from P / distance of closest boundary point from P. Find the point that minimizes this ratio.(Instead of ratio one can minimize the difference between these two quantities and find another center).

3. For each P in the interior of T, consider the average over the orientation angle of all chords of T that pass thru P and find the point for which this average chord length is maximum.(thie average can be computed over all points on the boundary of T and then, the center could be different).

Numerical experiments indicate that for a general triangle, none of these triangle centers necessarily coincides with any of the centroid, orthocenter or incenter. I don't know if these centers have been studied. I don't know if any of these centers always coincides with any other known and documented triangle center - am no expert in using resources such as the encyclopedia of triangle centers. So, let me for now merely note these questions.