Talk:Isoperimetric inequality

From Wikipedia, the free encyclopedia
Jump to: navigation, search
WikiProject Mathematics (Rated Start-class, Mid-priority)
WikiProject Mathematics
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
Start Class
Mid Priority
 Field: Geometry

It seems to me that the "proof using standard calculus" is flawed. The second term cannot be varied indepently of the first. Asides, it is necessary to fix the perimeter of the curve; otherwise we can make the area infinite by blowing up the figure. This boundary condition is not included at all in the "proof".

Yes, quite so. In fact it would be better to replace this so-called proof by (for instance) the proof from Hardy, Littlewood and Polya's book "Inequalities": they show that this isoperimetric inequality can be derived from Wirtinger's inequality (already proved in Wikipedia on its own page). Fathead99 08:44, 11 June 2007 (UTC)

I was coming to the talk page to make the same point. I will move the flawed proof to the talk page so it can be scruitinized, and perhaps someone might fix it, of write i one of the proofs aboveBilllion 21:42, 4 July 2007 (UTC)

Flawed "proof" using standard results[edit]

A proof of the isoperimetric theorem in 2D using the standard results for the length and area of a polar curve is as follows. Consider a shape described by the polar equation

r=f(\theta),\, f(0)=f(2\pi).\,\!

Consider a segment of this shape between arbitrary angles, θ1 and θ.

Let A be segment's area and s be segment's arc length. Then:

A=\frac{1}{2}\int_{\theta_1}^{\theta} r^2 d\theta\,\!
s=\int_{\theta_1}^{\theta} \sqrt{r^2 + \left (\frac{dr}{d\theta}\right )^2}\, d\theta,\,\!
\therefore s=\int_{\theta_1}^{\theta} \sqrt{2 \left (\frac{dA}{d\theta}\right ) + \left (\frac{dr}{d\theta}\right )^2}\, d\theta,\,\!
\therefore\left (\frac{ds}{d\theta}\right )^2 = 2\left (\frac{dA}{d\theta}\right ) + \left (\frac{dr}{d\theta}\right )^2,\,\!
\therefore \frac{dA}{d\theta}=\frac{1}{2}\left [\left (\frac{ds}{d\theta}\right )^2-\left (\frac{dr}{d\theta}\right )^2\right ],
\therefore A = \frac{1}{2}\int_{\theta_1}^{\theta}\left (\frac{ds}{d\theta}\right )^2\, d\theta - \frac{1}{2}\int_{\theta_1}^{\theta}\left (\frac{dr}{d\theta}\right )^2\, d\theta.\,\!

Since both of these terms contain squares and hence must be positive, A is maximised when the derivative dr/dθ = 0. It follows that r is a constant, QED.

Dido's problem[edit]

The article uses the term Dido's problem. And the lead-in mentions that varieties of this problem have roots in antiquity. Should these two matters and their connection (if any) be briefly explicated further? -- Cimon Avaro; on a pogostick. (talk) 15:47, 14 December 2007 (UTC)

Lagrange's contribution[edit]

I've removed the following text from the article:

Joseph Louis Lagrange solved the isoperimetric problem in 1755 under the assumption that the bounding curve be continuously differentiable, when he was only 19.[1]

  1. ^ W. W. Rouse Ball, "A Short Account of the History of Mathematics" (Fourth Edition, 1908)

At best, Lagrange has given a variational argument that circle is a local maximum of the area functional under the fixed length constraint (cf Lagrange multipliers). Not only is smoothness assumed, but also the proof along these lines leaves open a possibility that a global maximum is not attained (which happens for some surfaces). Weierstrass was the first to clearly enunciate and overcome this difficulty. As a side remark, copying material from one Wiki-article to another without verification has the danger of unwittingly spreading inaccurate information. Arcfrk (talk) 20:47, 11 February 2008 (UTC)

Several comments
* The threshold for inclusion in Wikipedia is verifiability, not truth. The Lagrange article appears to have a reliable source saying that he solved this problem, therefore it is not at all out of line to include it in this article. If the claim that Lagrange solved this problem is widely known, and my search on google seems to show that it might be, then WP:NPOV also says it is reasonable to talk about it here. WP:NPOV does not require every minor point of view to be covered though and I'm not an expert in this area so I really can't judge this.
* I think this article could be improved by giving more history behind the partial solutions. The article mentions that "Although the circle appears to be an obvious solution to the problem, proving this fact is rather difficult.". The explanation you have given above helps explain why this is hard to prove. Rather than deleting the mention of Lagrange, I think the article would be improved by adding your comments. Other "wrong" or "incomplete" solutions would also be interesting to read about, in my opinion.
* If Lagrange really didn't not have much of an impact on the isoperimetry problem, then I think it would be appropriate to edit that article too. The particular reference I cited seems to be heavily used by the Lagrange article, so if it is not a reliable source, then it shouldn't be used there either.
If nothing else, I've learned more about the history of this math problem, so I appreciate the education. Wrs1864 (talk) —Preceding comment was added at 21:11, 11 February 2008 (UTC)
Arcfrk is quite right, and I really should have realized this in my attempt to correct the text. The difficulty is that the Lagrangian for the problem is non-convex (as Arcfrk points out, this is the case of interest for Lagrange multipliers). In particular, existence of an extremum is by no means guaranteed a priori. What might be said is that Lagrange "found the extremals" of the associated variational problem. However, this probably misleadingly overstates the significance of Lagrange's contribution to the problem (and may have been the original source of the mixup). I think Lagrange should not be mentioned here after all, and the statement should be removed from his biographical article as well. Silly rabbit (talk) 22:16, 11 February 2008 (UTC)
The history section here is about 5% complete, so obviously it needs a lot of work. I've edited the Lagrange biography a bit, but unfortunately, it had originally been lifted verbatim from Rouse Ball's text, and contains tons of misleading statements in flowery language. Worse, some of the subsequent editors tried to "clarify" the cryptic turns of phrase by expanding them into mathematical claims, not always correctly. Notwithstanding the popularity of Ball's writings, he cannot be considered a reliable source from the academic point of view of history of mathematics. Arcfrk (talk) 22:58, 11 February 2008 (UTC)


If I recall, Pappus made important early contributions to isoperimetry, and should probably be mentioned here. Does anyone have a good reference? I should be able to dig one up in the next few days. Silly rabbit (talk) 22:18, 11 February 2008 (UTC)

There are some references in the website linked at the bottom. I am still searching for a good single source giving an overview of the history of classical isoperimetric problem through early 20th century. Blaschke's "Kreis und Kügel", Bonnessen–Fenchel, Hadwiger, Burago–Zalgaller and other standard mathematical sources tell only a small part of the story, making tantalizing hints (because it was, presumably, too well known to give the full account?) Arcfrk (talk) 22:49, 11 February 2008 (UTC)
Chavel's book on differential geometry is probably the best source. He may have spent more time than anyone else thinking about different proofs of the isoperimetric inequality. Katzmik (talk) 14:21, 1 June 2008 (UTC)

Spherical isoperimetric inequality[edit]

Presumably the statement currently on the page, that L^2 <= A(4pi-A) is the wrong way round, and it should say A(4pi-A) <= L^2 ? Fathead99 (talk) 09:27, 5 March 2010 (UTC)

Good catch! Fixed. Arcfrk (talk) 00:03, 6 March 2010 (UTC)