This question comes from an exam about 15 years ago.
How to find the maximal volume of a rectangular parallelepiped inscribed in an ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$?
I think this should be solved by Lagrange mulitpliers. but it is not given that the parallelepiped is parallel to the axes, so I cannot apply the method immediately.
I tired to prove that the only parallelepiped in an ellipsoid must be parallel to the axes; even though this seems obvious, I found no way of proving it rigorously.
If I can prove this, then the ensuing steps by Lagrange are not so difficult to me.
So any help is well-appreciated.
Best Answer
Apply a linear transformation with det=1 that turns yr ellipsoid into a sphere. It transforms the set of parallelepipeds inscribed in the ellipsoid into the set of parallelepipeds inscribed in the sphere, preserving their volumes. Show that the parallelepiped of maximum volume inscribed in the sphere is a cube. Transform back to the ellipsoid.