Talk:Isoperimetric inequality

Contents formatting error

The 'in R^n' section in the contents appears as https://wiki.riteme.site/wiki/Isoperimetric_inequality#In_%7F'%22%60UNIQ--postMath-00000019-QINU%60%22'%7F which probably isn't desired. I don't know how to embed math in contents so can't fix sadly. — Preceding unsigned comment added by Hu5k3rDu (talk • contribs) 10:39, 9 February 2022 (UTC)[reply]

Untitled

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)[reply]

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)[reply]

Flawed "proof" using standard results

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

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)[reply]

Lagrange's contribution

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]

^ 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)[reply]

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)[reply]

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)[reply]

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)[reply]

Pappus

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)[reply]

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)[reply]

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)[reply]

Spherical isoperimetric inequality

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)[reply]

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

Assessment comment

The comment(s) below were originally left at Talk:Isoperimetric inequality/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

This is a very important topic in geometry, but the treatment at the moment is very eclectic. Arcfrk (talk) 05:02, 9 February 2008 (UTC)[reply]

Substituted at 18:23, 17 July 2016 (UTC)

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Was the basic problem really hard to prove?

"Although the circle appears to be an obvious solution to the problem, proving this fact is rather difficult."

But... Really? Stating about something true AND obvious that it was hard to prove rigorously kind of gives intuition a win over serious mathematics (even if other obvious intuitions are sometimes wrong). It is the kind of idea that flat-earthers and mystics would like to astound people with. "Look, how hard it was for the main-stream so-called scientists to understand this simple fact!"

I don't advocate for fabricating a more "flattering" and less abusable story, but I do hope that experts: 1. Make sure that claims that something has been hard to prove are also well founded, i.e. not contaminated by popular science misconceptions ("urban legends"), fringe sources etc., or – better – that the difficulties are explained: "The obvious intuitive approach X does not work because of counterexamples / inappropriate problem splitting / difficulty formalizing it / missing consensus over necessary underlying theories / something else." 2. Also make sure that claims about harder-to-grasp generalizations are not transferred unchecked to the simplest versions of the problem.

The intuitive idea of "flipping" dents to increase area will not always work, as a flipped dent may intersect another part of the non-convex shape. It is, however, quite obvious to me that the convex hull of any non-convex shape will have a larger area and a smaller or equal squared perimeter. Uniform scaling of the resulting shape allows, without altering L^2/A, preservation of the optimization constraint (area for perimeter minimization or perimeter for area maximization). I am not a mathematician, and what seems obvious to me may not be easy to prove, or even necessarily true. But these are the kind of intuitive approaches I would like to see being proposed and challenged. Elias (talk) 12:31, 9 November 2021 (UTC)[reply]

Contradictory sections?

Is this possibly contradictory?

Compare [lead section]

In $n$ -dimensional space $\mathbb {R} ^{n}$ the inequality lower bounds the surface area or perimeter $\operatorname {per} (S)$ of a set $S\subset \mathbb {R} ^{n}$ by its volume $\operatorname {vol} (S)$ ,

\operatorname {per} (S)\geq n\operatorname {vol} (S)^{\frac {n-1}{n}}\,\operatorname {vol} (B_{1})^{\frac {1}{n}}

,

where $B_{1}\subset \mathbb {R} ^{n}$ is a unit sphere. The equality holds only when $S$ is a sphere in $\mathbb {R} ^{n}$ .

to [In $\mathbb {R} ^{n}$ section]

Given a bounded set $S\subset \mathbb {R} ^{n}$ with surface area $\operatorname {per} (S)$ and volume $\operatorname {vol} (S)$ , the isoperimetric inequality states

\operatorname {per} (S)\geq n\operatorname {vol} (S)^{\frac {n-1}{n}}\,\operatorname {vol} (B_{1})^{\frac {1}{n}}

,

where $B_{1}\subset \mathbb {R} ^{n}$ is a unit ball. The equality holds when $S$ is a ball in $\mathbb {R} ^{n}$ . Under additional restrictions on the set (such as convexity, regularity, smooth boundary), the equality holds for a ball only.

Additional conditions seem to be required (convexity, regularity, smooth boundary etc.) for making the sphere/ball special. Yet, these requirements are not in the lead section. And the requirements aren't even properly listed, they're represented in a such-as style. A1E6 (talk) 22:04, 10 November 2021 (UTC)[reply]

Round Sphere

In: "Since the amount of water in a drop is fixed, surface tension forces the drop into a shape which minimizes the surface area of the drop, namely a round sphere.", "...round sphere" is redundant, isn't it? Shouldn't it be changed to just "sphere"? 174.194.148.3 (talk) 15:54, 12 August 2022 (UTC)[reply]

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

[1]