Definition: The Hausdorff distance between two point sets is the greatest of all the distances from a point in one set to the closest point in the other set.
Question: Given a general triangle T, to find and characterize the ellipse E that minimizes the Hausdorff distance from T. It is not clear to me if E is unique.
Note 1: Some further questions/thoughts are recorded at https://nandacumar.blogspot.com/2023/01/approximating-triangles-by-ellipses.html and https://nandacumar.blogspot.com/2023/01/approximating-planar-convex-sets-by-n.html
Note 2: As has been pointed out by Matt F in comments below, the special case of the ellipse being a circle is itself of nontrivial interest.