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.
Best Answer
A classical result along these lines is Bonnesen's 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 $C = \pi^2/4$.