Maximin diameter of a transformed convex figure

Question

You are given a compact convex figure in the plane, $C$.

Your goal is to transform $C$ to a different figure $C'$ with the smallest possible diameter, as long as:

• The trasformation from $C$ to $C'$ is affine and invertible (e.g. scaling in one or more directions);
• $C'$ contains a disc with diameter $1$.

For example:

• If $C$ is an ellipse, you can transform it to a disc with diameter $1$.
• If $C$ is a parallelogram, you can transform it to a square with side-length $1$ and diameter $\sqrt{2}$.
• If $C$ is a triangle, you can transform it to an equilateral triangle with diameter $\sqrt{3}$.

An ellipse is obviously the best case; what is the worst case?