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:

- For a general ellipse $E\subset T \subset D$ centred at $(c, 0)$, what is the maximum area ellipse?
- 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 $(c, 0, 0)$?
- 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.)
- 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:

- $E$ 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 $t_y=\frac{1}{\sqrt{8}}\sqrt{1-4c^2+\sqrt{1+8c^2}}$.
- The radius of the largest spherical $E$ is $r=\frac{1}{3}(\sqrt{4-3c^2}-1)$.
- $E$ 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 $t_y=t_z=\frac{1}{\sqrt{18}}\sqrt{1-3c^2+\sqrt{1+3c^2}}$.
- $E$ 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!