Is the following inequality known? I believe it's true, but I could find no reference.

For any convex body $C$ in the plane we have $$\left(4-\frac{8}{\pi}\right)area(C)\leq > diam(C)(per(C)-2diam(C)).$$

If true, this would be tight, with equality when $C$ is a disk. If it turns out not to be true, then it still makes sense to look for the best constant with the $area(C)$ term.

The following similar inequality $diam(C)(per(C)-2diam(C))\leq\frac{4}{\sqrt3}area(C)$ (equality when $C$ is an equilateral triangle) is definitely known and is proven here.

Journal of Inequalities in Pure and Applied Mathematics, might help, although I don't see your inequality there (but maybe I missed it amidst the welter of notation). PDF download: emis.de/journals/JIPAM/images/016_99_JIPAM/016_99.pdf – Joseph O'Rourke Feb 1 '13 at 16:53