# Generalising Euler's formula to ellipses and three dimensions

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!

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This may be related to the Steiner inellipse of a triangle (en.wikipedia.org/wiki/Steiner_inellipse) or, in higher dimensions, the John ellipsoid (en.wikipedia.org/wiki/John_ellipsoid). –  Gerry Myerson Oct 10 '13 at 3:08
This was actually my first thought, but I think it can be used only to solve a slightly different question. The Steiner inellipse is centred on the centroid of $T$ and has area proportional to the area of $T$. So construct the largest area $T$ with centroid $(c,0)$ and circumcircle $D$. The resulting inellipse has area $\frac{\pi}{4}(1-c)^\frac{3}{2}\sqrt{1+3c}$, which is smaller than my solution to question 1. In fact, my solution appears to be somewhere between the Steiner inellipse and the circle solution. The largest area ellipse I have found is not the Steiner inellipse of $T\subset\ D$. –  Antony Oct 10 '13 at 10:38
Just to clarify, when I say "my solution appears to be somewhere between the Steiner inellipse and the circle solution", I am talking about the semiaxes rather than the area. The Steiner inellipse solution and mine are both oriented with the minor axis along the $x$ axis. We have $s_x\leq t_x\leq r$ and $r\leq t_y \leq s_y$, where $r=\frac{1}{2}(1-c^2)$ and $s_{x,y}$ are the Steiner inellipse semiaxes. –  Antony Oct 10 '13 at 11:51