With no knowledge of calculus of variation, is it possible to prove that among closed curves with fixed length in 2D, a circle has the most area?

Question

This problem is talked about in the topic of calculus of variation. For someone who is not familiar with this area of math, how can we prove it?

I thought this was equivalent to proving that among the shapes with fixed area, a circle has the least perimeter.

Then I thought that we could consider a circle, then consider an infinitesimal arc of it subtending an infinitesimal angle. Then we could say that if you try to alter the arc at that point, the perimeter will get larger. But the area will not change much (because the infinitesimal rank of area is 2 but that of a line segment is 1). So it means that circle must have the least perimeter.

But then I came up with the conclusion that this can be applied to any kind of curve (more precisely, not just circle). So this solution will not work.

Is it possible? Any ideas?

Answer 1

I heard something about the following proof. But it contains some hand waving. Maybe you can argue the details better.

Proof.

Consider any 2D-shape $S$ (preferably convex or at least without holes). Put some line $\ell$ through the center of the shape. If the shape is not symmetric w.r.t. the line $\ell$, then you can build a symmetrized shape $S'$ from it (imagine this process like symmetrizing a stack of books on a table, each layer gets moved to be centered over the middle axis, or see cavalerie's principle). This shape will have the same area but will have less perimeter (here, I have no clue how to argue, but it was told to be as if it were obvious).

So this means as long as there is a line through the center of $S$ that is not a symmetry axis of $S$, there will be a shape $S'$ with better area perimeter ratio. So the best we can do is to choose a shape that is symmetric w.r.t. to all axis trough its center. And this is a circle. $\square$

Answer 2

There are numerous proofs of the isoperimetric inequality and many nice surveys that discuss such proofs. One such survey is by Viktor Blasjo (with additional diacritic symbols, sorry):

Blåsjö, Viktor. The isoperimetric problem. Amer. Math. Monthly 112 (2005), no. 6, 526–566.

See here.

Answer 3

Some minimal surface physical models give an appealing visual proof and also contain the variational calculus (CV) principle of maximization with relevant maths inside of them.

When a round rigid circle loop with a string of length more than its diameter is tied and dipped in a soap solution and taken out, it forms a film disc, the string makes arbitrary excursions on the film.

Next when a portion (red) is pin-pricked to burst it the blank area so formed has maximum area for given string length, being pulled by forces of surface tension.

The same Lagrangian applies for the two cases: given length has max area or given area is surrounded by minimum string length.

If such things could be proved without CV, why was it brought in as technique?