Give an example of a non-ellipsoid ovalid The compact, connected surfaces of $\mathbb{R}^3$ for which the Gaussian curvature is always positive are called ovaloids.

So, ellipsoids are ovaloids.

Is there an ovaloid that is not an ellipsoid? Some literature?
How about the surface with equation $x^4+y^4+z^4=1$?
I happen to have been acquainted with Robert Greene from my days at ucla... from geometry seminars etc...  and I happen to remember there is a paper by him, entitled "ovaloids" or something like that,  and the reference is in the bibliography of one of the volumes of Spivak's comprehensive introduction to differential geometry.

Perhaps look it up...  (it's back there somewhere... )