Maximum ratio between diameter of shape and diameter of enclosing circle For a 2-dimensional closed convex shape $C$, define:


*

*$d(C)$ = the diameter of $C$ (the largest distance between two points in $C$).

*$D(C)$ = the diameter of the smallest circle containing $C$.


Often $d(C)=D(C)$, for example in a unit square they are both $\sqrt 2$.
Sometimes $d(C)<D(C)$, for example in a unit equilateral triangle $d(C)=1$ and $D(C)=2/\sqrt 3$.
MY QUESTION: What is the largest value of $D(C)/d(C)$? (Can it be larger than $2/\sqrt 3$?)

I think $2/\sqrt 3 $ is the largest possible value: For simplicity only consider $C$ to be a convex polygon. If $C$ is a line, then the ratio is $1$. 
Now think about triangles: If we fix the diameter of the enclosing circle, then the triangle that produces the largest ratio (i.e. the smallest $d(C)$) is equilateral. 
If we still fix $D(C)$ and try to construct a polygon $C$ in this circle that has a very small diameter, we can always embed a triangle into $C$, which has the same diameter as $C$. However, this diameter is greater or equal to the diameter of an equilateral triangle with the same enclosing circle. Hence, the ratio $D(C)/d(C)$ is smaller or equal to the ratio of an equilateral triangle. 
Since arbitrary convex shapes can be approximated by polygons, the claim follows. 
