The lower bound of the ratio of the diameter of a convex polygon to its perimeter

Question

Let's say I have a convex polygon $P$. $D$ is its diameter (the distance between its two farthest-apart vertices) and $\varphi$ is its perimeter. I believe the following to be true:

$$D/\varphi \ge 1/\pi$$

Could someone prove this or provide a counterexample?

Answer 1

Yes, this is true, see Table 2.1 in the paper Inequalities for Convex Sets by Scott and Awyong. See also https://mathoverflow.net/questions/329845/inequality-fraccd-max-le-pi-relating-perimeter-and-diameter-of-plana and the references therein.