On Elliptical Containers

The problem:

Given a convex 2D region C, find the ellipse of least perimeter containing C.

Special case: Experimentally, we saw that even for a given triangular region, the minimal perimeter elliptical container is not in general the same as the minimal area elliptical container (the latter is of course, the well studied Steiner ellipse).

Is something known about the center and orientation of the least perimeter containing ellipse for a given triangle? And what about such an elliptical container for a general convex region? And of course, the question can be generalized to higher dimensions.

There is a vague hope that despite the absence of a closed form expression for ellipse perimeter, one could find the dimensions of the containing ellipse with minimum perimeter. For example, one can compare the perimeters of two equal area ellipses without knowing the actual perimeters by just looking at the major and minor axes; so....

Note: It is not hard to see that if the container is a rectangle, the least area rectangle container and least perimeter rectangle container of a given convex region need not be the same (for an example, consider the "square being trimmed" example given in last post). It looks conceivable (not sure though!) that both such rectangular containers can be found by the rotating calipers method. However both the least area and least perimeter elliptical containers need not be aligned with any of the edges of the convex region C so a variant of such an approach might not work.

However, here is a bit of guesswork: For any fixed convex region C and an angle alpha measured w r to any fixed reference direction, we have a unique ellipse of minimal perimeter aligned at angle alpha and containing C. As C is kept fixed and alpha varied, the parameters of the minimal perimeter containing ellipse aligned with alpha - {its center, the axes a and b, its perimeter} - all appear to change continuously. And the number of times the derivative w r to alpha of the perimeter changes sign appears limited by the number of sides in the convex region. So a well-defined global minimum of this perimeter should exist.

Note added on December 3rd, 2016:

Experiments were done to guess the center or the foci of the least perimeter elliptical containers of isosceles triangles. All findings so far are negative. The center of the least perimeter elliptical container of a triangle is in general NOT the centroid, NOT the orthocenter,... The foci of this container also could not be pinned to any of the known special points of the triangle. These are only numerical observations.

Wrong Conjectures

Recording two conjectures and counter-examples.... Thanks to Prof. Erich Friedman.

1. Given any convex region C. Let R be the largest area rectangle that can be drawn inside C and R' the smallest area rectangle that can be drawn containing C. Claim: R and R' necessarily have the same alignment (same axes of symmetry). Note: A wider version of this claim could be stated generalizing from rectangles to any other convex shape with reflection symmetry.

Counter-example: Take a square, and shave off opposite corners at a 45 degree angle. For small cuts, the exterior square and interior square are optimal. For large cuts, a 45 degree rectangle is better. It would be a big surprise if the two phase transitions took place at exactly the same point.

And indeed, as can be checked numerically, the two phase transitions DO NOT happen at the same point.

o 2. Claim: The rectangle with largest perimeter that can be drawn inside any convex C is either degenerate (zero area, only length) or the same as the interior rectangle with highest area. Similarly, from among the rectangles containing C, the one with the least perimeter is the same as the one with least area. Note: This claim could be generalized from rectangles to ellipses or any other regions with reflection symmetry.

Counter: This claim is easily shown to fail by considering rectangles inscribed in any given ellipse. The inscribed rectangle with max perimeter is not usually the one with most area.

As for max containing rectangles, by considering the counterexample for claim 1( the progressively shaved square), we see that that for a certain stage in the shaving, the the 45 degree outer rectangle has less area but the outer square has less perimeter.

Note: Also tried to find CONTAINING ellipses with least area and least perimeter for a family of isosceles triangles all with same fixed apex and altitude but different bases. The least area containing ellipses of these triangles all have the same center (the centroid, which they all share anyways) - indeed these are their Steiner ellipses - but the centers of the least perimeter containing ellipses vary.