I have been searching for a version of the isoperimetric inequality which is something like:


$P(\Omega) - 2\sqrt{\pi} Vol(\Omega)^{1/2} \geq \pi (r\_{out}^2 - r\_{in}^2)$ where $r\_{out}$ and $r\_{in}$ are the inner and outer radius of a given set. There are of course details which I am missing such as what kind of sets this applies to (clearly connected and possibly simply connected). I was hoping somebody may recognize this inequality and be able to direct me to a source for it along with a proof.


**Update:** I'm curious if anyone can direct me to a some papers which relate the isoperimetric deficit to Hausdorff distance. Such as:
$P(\Omega)^2 - 4\pi Vol(\Omega) \geq C d\_H(\Omega,B)^2$ whre $B$ is a sphere in $\mathbb{R}^2$
which may be the inner or outer circle.


**Update April 12:** I would like to know if the first Bonnesen inequality written below is strictly stronger than the one in higher dimensions? In particular, if one considers the Fraenkel assymetry $\alpha(\Omega) = \min\_B |\Omega \Delta B|$ where $|B|=|\Omega|$, does it hold on a bounded domain that


$ r\_{out}^2 - r\_{in}^2 \leq C \alpha(\Omega)$,


for some constant $C>0$? This seems like it should be true but I can't seem to find a concise proof of it.



Stronger version of the isoperimetric inequality
A classical result along these lines is [Bonnesen's inequality](http://en.wikipedia.org/wiki/Bonnesen%27s_inequality), which states
$$
L^2 - 4\pi A \ge \pi^2 (r\_{out} - r\_{in})^2,
$$
where $L$ is the length and $A$ is the enclosed area of a simple planar closed curve. There are many other results along these lines, called "stability estimates" for the isoperimetric inequality. There are several pointers to the literature in Note 6 following section 6.2 of Schneider's book *Convex Bodies: the Brunn-Minkowski Theory*.


**Added:** Bonnesen's inequality also suffices for the updated question. If $B\_{in}$ and $B\_{out}$ are the inner and outer disks, respectively, then since $B\_{in} \subseteq \Omega \subseteq B\_{out}$,
$$
d\_H(\Omega, B) \le d\_H(B\_{in},B\_{out}) \le 2r\_{out} - 2r\_{in}
$$
(the extreme case in the latter inequality being when the circles are tangent), so you get your desired result with $C = \pi^2/4$.


