another diameter-perimeter-area inequality

Recently I learnt that $$\inf\frac{diam(C)(per(C)-2diam(C))}{area(C)}=0$$ 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 $$diam(C)(per(C)-2diam(C))\geq c\cdot area(C),$$ 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 $per(C)\geq2diam(C)$ 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 $\epsilon$)?

-