[John's Theorem](http://en.wikipedia.org/wiki/John_ellipsoid) can be stated as "To every compact, convex body, there is a unique inscribed ellipsoid, whose volume is maximal among all inscribed ellipsoids." It goes on to classify this maximal ellipsoid. By using this theorem, one can prove that the ellipsoid of maximal volume which is contained in a square is a circle. 


This strikes me as a problem which was probably studied well before Fritz John, and yet I have been unable to prove the statement about squares and circles in an elegant, but low-brow manner. Any thoughts?



Maximal Ellipsoid
Here's an attempt at a low-brow proof. Take a max area ellipse. Apply an affine transform to
make it a circle; then the problem becomes to show that a minimal area parallelogram containing a circle is a square. It is easy to see that both the height of the parallelogram
and its base are at least the diameter. Q.E.D.


