n-ellipse questions

The following are queries put up last week at mathorverflow:
-----------
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.
-----------
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.