User matt pusey - MathOverflow

Answer by Matt Pusey for Does every ellipse inside a tetrahedron inside a ball fit in a triangle inside the ball?

2012-10-08T15:02:40Z

Mihai-Dorian Vidrighin has suggested the following idea. (He's asked me to post it because he can't post images.)

For simplicity, assume that the setup is "tight" in that $E$ meets every face of $T$ and the vertices of $T$ are on the boundary of $B$. The generalisation should be straightforward.

If any vertices of $T$ lie in the plane of $E$ then the cross-section is already a triangle, so assume otherwise. Pick an arbitrary vertex of $T$ and draw a cone staring from there with $E$ (shown in red) as a base. This cone meets the opposite face of $T$ in a new ellipse $E'$ (shown as a dotted black curve):

$E'$ lies inside a triangle (the face of $T$), which itself lies inside a circle (the cross-section of $B$). By our simplifying assumption, $E'$ touches the edges of the triangle and the triangle's vertices are on the circle. Therefore Poncelet's Porism applies. Hence we can find a different triangle around $E'$ with one edge in the plane of $E$.

Define a new tetrahedron (shown in green) using the chosen vertex of $T$ and the new triangle. By construction it still contains the cone identified before, and in particular it still contains $E$. But the cross-section in the plane of $E$ is now a triangle (shown in thick black).

Does every ellipse inside a tetrahedron inside a ball fit in a triangle inside the ball?

2012-09-28T11:16:49Z

In three-dimensional euclidean space, consider the closed unit ball $B$. Let $T$ be a tetrahedron, and $E$ an ellipse, with $E \subset T \subset B$. Does there necessarily exist a triangle $T'$ with $E \subset T' \subset B$?

Clearly if 1 vertex of the tetrahedron is on one side of the plane of of the ellipse, and the other 3 vertices are on the other side, then intersecting the plane with the tetrahedron gives such a triangle. The interesting case is when 2 vertices of the tetrahedron are on each side of the plane.

I work in quantum information theory, and have come up with a conjecture that, remarkably, is true if and only if the answer to the above question is "yes"! Since this is very far from the sort of thing I normally think about, I don't even know where to begin to look in the mathematics literature, so even just a pointer would be a big help.

Comment by Matt Pusey

2012-10-10T17:20:20Z

Mihai has pointed out that his proof includes a partial version of a Poncolet's Porism for ellipsoids inside tetrahedron inside ellipsoids, it might be interesting to see how far that can go. My original question could also be generalised to simplices in higher dimensions. But I think my original problem is solved (the best way to reduce the none-tight case is perhaps to shrink the base of the tetrahedron to make it tight, then shrink the cross-section of the Bloch sphere to a tight ellipse round that.) Thanks for all the fantastic suggestions, and in particular for the name of the Porism.

Comment by Matt Pusey

2012-09-28T14:33:16Z

Thanks. I agree that that ellipse looks like it won't fit in a triangle, but I don't agree that it fits inside the tetrahedron. If you try to make a thin tetrahedron like that then a typical cross-section through it will be a diamond which your ellipse wouldn't fit into. If you have Mathematica, the code at pastebin.com/UCAYTPpc shows the sort of tetrahedron I think your talking about and lets you look at cross-sections through it.