# Maximal tetrahedra inscribed in ellipsoid

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:

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Something maybe related: Oded Schramm's "How to Cage an Egg" proves that there is a 6 dimensional family of convex tetrahedra which "midscribe" an ellipsoid (that is, all its edges are tangent to the ellipsoid). digizeitschriften.de/dms/img/?PPN=GDZPPN002109468 –  j.c. Oct 19 '11 at 21:35
@jc: Very cool! I didn't know about this theorem. It appears to be connected to "Thurston's proof that the sphere can be midscribed by a polytope of given combinatorics." –  Joseph O'Rourke Oct 19 '11 at 23:52
I'm not surprised that ellipses don't have this property; it's good to have Henry's counterexample. Two comments: First, for the sphere (a special case), there is a 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. –  Robert Bryant Oct 20 '11 at 20:21
@Robert: Nice question re sphere & 3-parameter family! –  Joseph O'Rourke Oct 20 '11 at 21:36

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.