Pietro Majer quoted the theorem of Michel Chasles in his MO question,
"Convex curves with many inscribed triangles maximizing perimeter,"
which states that the triangles of maximum perimeter inscribed in an ellipse form a one-parameter family, all "billiard triangles."
(This theorem is a special case of Theorem 17.6.6 in Berger's
Geometry II,
p.243, which establishes the result for convex $n$-gons.)
My question is whether or not any generalization to higher dimensions is known, and in particular,
whether there is an analogous theorem for tetrahedra inscribed in ellipsoids in $\mathbb{R}^3$.
Presumably the generalization would state that there is a two-parameter family of
maximum surface-area tetrahedra inscribed in an ellipsoid.
I have not found such a theorem, and would be interested to learn whether or not it (or
some variant) is known.

**Update**. Here is Henry Cohn's example, just showing the $z=0$ ellipse slice
that contains his degenerate maximal tetrahedra:

tangentto the ellipsoid). digizeitschriften.de/dms/img/?PPN=GDZPPN002109468 $\endgroup$three-parameter family of maximal surface-area tetrahedra inscribed in the sphere, namely the regular ones. Thus, the first question would seem to be "Are there any other convex bodies besides the sphere that support a $3$-parameter family?". Second, an analysis similar to the one for ellipses in the plane indicates that there may well be many other surfaces that support a $2$-parameter family. Details follow when time permits. $\endgroup$