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](http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1929_3_46_/ASENS_1929_3_46__345_0/ASENS_1929_3_46__345_0.pdf). 



a diameter-perimeter-area inequality for convex figures
This inequality is not true. Consider the rectangle on $\mathbb R^2$ with vertices $(\pm 1, 0)$, $(0, \pm \varepsilon)$. Then on the left you have $2\varepsilon(4-8/\pi)$ on the right you have approximatively $4\varepsilon^2$.


