You are given a convex shape $S$ in the plane. You are allowed to apply any affine transformation to $S$. Then, you have to pick an axes-parallel square contained in (the transformed) $S$, and an axes-parallel square containing $S$, such that the ratio between the side-lengths of the squares is as small as possible. What is the smallest ratio that can be attained for every $S$?
Examples:
- If $S$ is a rectangle, then it can be transformed into an axes-parallel square, so the ratio is $1$.
- If $S$ is an ellipse, then it can be transformed into a circle, so the ratio is $\sqrt{2}$.
- If $S$ is a triangle, then it can be transformed into an isosceles right-angled triangle, so the ratio is $2$.
Apparently, a triangle is the worst case, but I could not prove it. Is it always possible to attain a ratio of 2? What is the smallest ratio that can always be attained?