covering a rectangle with an ellipse of minimal area

Question

Given a rectangle $B$ I would like to find an ellipse $C$ such that $B$ is contained inside $C$. Of course this is always possible, but I was wondering can we do this in a way that the area of $C$ is controlled? More specifically, can we prove that there exists a constant $D>0$ such that given any $B$ there exists $C$ containing $B$ with $$ \frac{Area (C)}{Area(B)} < D? $$

Answer 1

The smallest ellipse containing $B$ touches the rectangle.

Scale the ellipse by $\alpha$ in the direction of an axis so that it becomes a circle $C'$. The rectangle remains a rectangle $B'$ with area scaled by $\alpha$ as well.

Now, the largest area of a rectangle in a circle is that of a square, namely $2r^2$. So $$\frac{Area(B)}{Area(C)}=\frac{\alpha Area(B')}{\alpha Area(C')}\ge\frac{2}{\pi}.$$ So if $\alpha$ is chosen to be the ratio of the sides of the rectangle, then $B'$ is a square, $C$ is an ellipse with ratio of major/minor axes of $\alpha$, and $$\frac{Area(C)}{Area(B)}=\frac{\pi}{2}$$ Hence take any $D\ge\frac{\pi}{2}$.