Stronger version of the isoperimetric inequality - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T04:01:44Z http://mathoverflow.net/feeds/question/55404 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/55404/stronger-version-of-the-isoperimetric-inequality Stronger version of the isoperimetric inequality Dorian 2011-02-14T12:44:06Z 2012-04-12T15:33:46Z <p>I have been searching for a version of the isoperimetric inequality which is something like:</p> <p>$P(\Omega) - 2\sqrt{\pi} Vol(\Omega)^{1/2} \geq \pi (r_{out}^2 - r_{in}^2)$ 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.</p> <p><strong>Update:</strong> 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.</p> <p><strong>Update April 12:</strong> 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</p> <p>$r_{out}^2 - r_{in}^2 \leq C \alpha(\Omega)$,</p> <p>for some constant $C>0$? This seems like it should be true but I can't seem to find a concise proof of it.</p> http://mathoverflow.net/questions/55404/stronger-version-of-the-isoperimetric-inequality/55411#55411 Answer by Andrey Rekalo for Stronger version of the isoperimetric inequality Andrey Rekalo 2011-02-14T13:44:08Z 2011-02-14T13:44:08Z <p>There is a sharpened version of the plane isoperimetric inequality due to Benson which involves the inner and outer radii. Let $$\Gamma=\{(r,\theta):\ r=r(s),\theta=\theta(s)\}$$ be a simple closed rectifiable curve on the plane, parametrized by the arc length $s$, and let $$r_1=\sup\{r:\ (r,\theta)\in\Gamma\},\qquad r_2= \inf\{r:\ (r,\theta)\in\Gamma\}.$$ Assume that $\Gamma$ winds once arround the inner circle. Then </p> <blockquote> <p>$$L^2-4\pi A\geq\frac{(2FA-2\pi E-\pi/(2F))^2}{1+4EF},$$</p> </blockquote> <p>where $L$ is the perimeter of $\Gamma$, $A$ is the area of the enclosed region, and $$F=\frac{1}{r_1-r_2},\qquad E=\frac{r_1r_2(r_1+r_2)}{(r_1-r_2)^2}.$$ </p> <p>The reference is: D.Benson, <a href="http://www.jstor.org/pss/2316850" rel="nofollow">"Sharpened Forms of the Plane Isoperimetric Inequality"</a>, <em>The American Mathematical Monthly</em>, Vol. 77 (1970), pp. 29-34.</p> http://mathoverflow.net/questions/55404/stronger-version-of-the-isoperimetric-inequality/55412#55412 Answer by Mark Meckes for Stronger version of the isoperimetric inequality Mark Meckes 2011-02-14T13:54:32Z 2011-04-28T12:17:31Z <p>A classical result along these lines is <a href="http://en.wikipedia.org/wiki/Bonnesen%27s_inequality" rel="nofollow">Bonnesen's inequality</a>, 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 <em>Convex Bodies: the Brunn-Minkowski Theory</em>.</p> <p><strong>Added:</strong> 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 $C = \pi^2/4$.</p> http://mathoverflow.net/questions/55404/stronger-version-of-the-isoperimetric-inequality/55443#55443 Answer by Gil Kalai for Stronger version of the isoperimetric inequality Gil Kalai 2011-02-14T20:41:29Z 2011-02-14T20:41:29Z <p>A very good source of Bonnesen type inequalties is the paper by Rovert Osserman entitled <a href="http://www.jstor.org/stable/2320297" rel="nofollow">Bonnesen style isoprimetric inequalities</a>, Americam Math Monthly 86(1979) 1-29. <a href="http://mathdl.maa.org/images/upload_library/22/Ford/RobertOsserman.pdf" rel="nofollow">Here is another link</a> for the same paper through <a href="http://mathdl.maa.org/mathDL/?pa=content&amp;sa=viewDocument&amp;nodeId=2956&amp;pf=1" rel="nofollow">this page</a>. Osserman's 1978 Bulletin AMS <a href="http://www.ams.org/journals/bull/1978-84-06/S0002-9904-1978-14553-4/S0002-9904-1978-14553-4.pdf" rel="nofollow">paper on the ioperimetric inequality</a> is also a good related source. </p>