We try to proceed from Least area and least perimeter triangles that contain a convex planar region - how different can they be?
The partial answer given to the above question shows a convex quadrilateral for which the least area triangle that contains it and the least perimeter triangle that contains the quad are different.
Definitions: Given a uniform planar convex region C, for any positive real $\alpha$, let us say the $\alpha$-moment of C is the least value of the integral of the product of the area of an element of C and $\alpha$-th power of the distance of this element from a fixed point. And the point from which the distances minimize the integral may be called the $\alpha$-center of C. Special case: the 2-moment of C is of course, its moment of inertia about an axis perpendicular to its plane and passing through its center of mass (the 2-center point).
Given a convex polygon P and a value of $\alpha$, how does one characterize and find that triangle containing P that has the least $\alpha$-moment?
Among all convex planar regions C of unit diameter, which one causes the $\alpha$-center to move the greatest distance if $\alpha$ is continuously changed from 0 to $\infty$? If C is centrally symmetric, the $\alpha$-center will not move at all. Guess: the shift of the $\alpha$-center is maximized when C is a triangle.
For a fixed convex planar region C, when $\alpha$ goes from 0 to $\infty$, how will the containing triangle with least $\alpha$-moment evolve? Will it, say, go from the least area containing triangle to the least perimeter containing triangle?
