TITLE: Maximal tetrahedra inscribed in ellipsoid
QUESTION [8 upvotes]: 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:

REPLY [6 votes]: I haven't proved anything, just done some numerical experiments, but I do not think there is always a two-parameter family of maximum-surface-area tetrahedra incribed in an ellipsoid (although you do get a two-parameter family of regular tetrahedra inscribed in a sphere).  For the ellipsoid defined by $x^2+2y^2+3z^2=1$, the maximum surface area appears to be $2\sqrt{2}$, and it is achieved by the one-parameter family of degenerate tetrahedra with vertices $\pm (\alpha,\sqrt{(1-\alpha^2)/2},0)$ and $\pm (-\sqrt{1-\alpha^2},\alpha/\sqrt{2},0)$ for $-1 \le \alpha \le 1$.  These are the only maximum-surface-area tetrahedra my program found for this ellipsoid.