Approximating planar convex sets by n-gons

Let me add a little bit to the last post here - on approximating triangles with ellipses (and viceversa)

In chapter 2 of 'Combinatorial Geometry', Pach and Agarwal discuss approximating convex compact planar sets by n-gons. They treat 2 cases: min area circumscribed n-gons and max area inscribed n-gons.

A third possible approximation is by n-gons that are closest to the given planar set in terms of hausdorff distance and that seems unexplored. And I don't even know about approximations by min perimeter circumscribed and max perimeter inscribed n-gons.

A bit more has been added in this post at mathoverflow:

(We can consider) other measures of distance between planar convex regions C1 and C2, for ex: the average least distance of a boundary point on C1 from a boundary point on C2. So, one could ask:

Question: Given a general triangle T, find that circle C (ellipse E) such that the average over boundary points of C(E) of closest distance from that point to a boundary point of T is minimized.