Let $D$ be the closed unit disk, $T$ a triangle and $E$ an ellipse with $E\subset T \subset D$. Without loss of generality say that $E$ is centred at Cartesian coordinates $(c, 0)$ with $0\leq c \leq 1$ being a variable parameter. The axes of $E$ are not necessarily aligned with the coordinate axes.


For the case that $E$ is a circle of radius $r$, we have the the Euler triangle formula that $r\leq\frac{1}{2}(1-c^2)$. This means that any circle $E\subset T \subset D$ has area $\leq \frac{\pi}{4}(1-c^2)^2$. I wish to generalise this result to find the maximum area ellipses and maximum volume ellipsoids with a given centre. In particular, my questions are:


1. For a general ellipse $E\subset T \subset D$ centred at $(c, 0)$, what is the maximum area ellipse?
2. In 3D, $B$ is the closed unit ball, $T$ is a tetrahedron and $E$ is an ellipsoid. What is the largest volume $E\subset T\subset B$ with $E$ being a sphere centred at $(c, 0, 0)$?
3. What is the largest volume ellipsoid $E\subset T\subset B$ with $E$ centred at $(c, 0, 0)$? (As with the 2D case, the ellipsoid axes are not necessarily aligned with the coordinate axes.)
4. What is the largest volume ellipsoid $E\subset B$ with $E$ centred at $(c, 0, 0)$? (No longer considering a tetrahedron.)


I believe I have the answers to all these questions worked out using a tool from quantum information theory (steering ellipsoids). However, I would like to know how easily the questions can be answered directly using just geometric considerations. My answers are:


1. $E$ has its axes aligned with the coordinate axes. The minor semiaxis is $t\_x=\frac{1}{4}(3-\sqrt{1+8c^2})$. The major semiaxis is $t\_y=\frac{1}{\sqrt{8}}\sqrt{1-4c^2+\sqrt{1+8c^2}}$.
2. The radius of the largest spherical $E$ is $r=\frac{1}{3}(\sqrt{4-3c^2}-1)$.
3. $E$ is an oblate spheroid with its axes aligned with the coordinates axes. The minor semiaxis is $t\_x=\frac{1}{3}(2-\sqrt{1+3c^2})$ and the major semiaxes are $t\_y=t\_z=\frac{1}{\sqrt{18}}\sqrt{1-3c^2+\sqrt{1+3c^2}}$.
4. $E$ is an oblate spheroid with its axes aligned with the coordinates axes. The minor semiaxis is $t\_x=1-c$ and the major semiaxes are $t\_y=t\_z=\sqrt{1-c}$.


Essentially, I am wondering whether these results are of any interest to geometers, i.e. have I found some interesting geometric results from physics-based arguments? I have found a few papers considering related questions (e.g. <http://math.sfsu.edu/federico/Articles/euler.pdf>‎) but nothing covering exactly these questions. Not being anything like a geometrist myself, I really have no idea whether these are difficult questions to answer geometrically.


Thanks!



Generalising Euler's formula to ellipses and three dimensions
I have finished working on these questions and written it up: [The Euler and Grace-Daniellson inequalities for nested triangles and tetrahedra: a derivation and generalisation using quantum information theory](http://arxiv.org/abs/1404.0525).


