Recently I learnt that $$\DeclareMathOperator{\diam}{diam}\DeclareMathOperator{\per}{per}\DeclareMathOperator{\area}{area} \inf\frac{\diam(C)(\per(C)-2\diam(C))}{\area(C)}=0$$ where the infimum is taken over all plane convex bodies $C$ (say, with non-zero area). In other words, there is no non-trivial (with $c>0$) inequality of the form $$\diam(C)(\per(C)-2\diam(C))\geq c\cdot \area(C),$$ that would hold for all $C$.

Now I'm wondering about inequalities of the form $$\diam(C)(\per(C)-(2-\epsilon)\diam(C))\geq c_{\epsilon}\cdot \area(C)$$ for $\epsilon>0$. Clearly, such an inequality is true for $c_\epsilon=\frac{4\epsilon}{\pi}$ (since $\per(C)\geq2\diam(C)$ and $\diam(C)^2\geq4/\pi\cdot \area(C)$).

The question is: ** what is the best $c_{\epsilon}$ for which the inequality holds?** When would equality be achieved (it might depend on $\epsilon$)?