Rectangular Parallelepipeds in Ellipsoids
J. Duncan, D. Khavinson, H. S. Shapiro · SIAM Review · 1996
In any general quadric with distinct semiaxes, the only inscribed rectangular parallelepipeds are the obvious ones—those which have their sides parallel to the principal axes. In a general ellipsoid with $n \geq 4$, no slice by a hyperplane can be a sphere.