## 2008年6月1日 星期日

### Investigating lengths and areas

Inspired by Kahoo, who wrote an article on some possible investigations that a primary school student can carry out of the Fibonacci sequence, let me also try posting some questions in geometry, so that some of you may try your hands on them. (Unfortunately I think the questions that follow will require knowledge from high school mathematics :p)

Suppose we are given a rope of length L.

1. What is the area of the largest rectangle that one can enclose using the given rope?

Do you have any intuition about this problem? Let's try some extreme cases. A rectangle of a given perimeter L is determined by its length, which can vary from 0 to L/2. Suppose we are in these extreme cases. What would the areas of the rectangles be? Suppose we gradually move from one extreme to another. What will happen to the area of the rectangle? Does that give you some intuition, and suggest how you could prove it?

Students who know one-variable calculus can solve this as a maximization problem; students who does not know calculus can solve it by some simple algebra (for instance, using the difference of squares formula, which is essentially the essence of the calculus method).

Graduate students in mathematics can also do this by flowing the rectangle into a square :p

2. What is the area of the largest triangle that one can enclose using the given rope?

A triangle with given perimeter is determined by 2 parameters (e.g. the length of two of its sides). So this would look like a two-dimensional maximization problem. Students who know multi-variable calculus can solve this by invoking Heron's formula. What if you don't know two-variable calculus? Is there a way of reducing the problem to one-variable?

The answer is yes, provided that you know the maximum exists in the first place: then look at the configuration for which the area of the triangle is a maximum, fixing one of the sides and vary the other two, one realizes (using one-variable calculus now, because the length of one of the variable sides determines the length of the other!) that the two variable sides have the same length. This would show indeed that the maximum can only be achieved when the lengths of all three sides of the triangle are the same, i.e. when it is an equilateral triangle.

Actually a simple geometric symmetry argument already leads to the above observation, and one can bypass calculus altogether!

3. What is the area of the largest shape that one can enclose using the given rope? (This is the famous isoperimetric inequality.)

First, what is the intuition?

Now that the shape of the enclosed area is arbitrary, we have an infinite degree of freedom when we maximize the area; indeed if one formulates the problem correctly, one can do a maximization involving only "countably many variables". For instance, check out the following two Fourier analytic proofs of the isoperimetric inequality in The everything seminar, that makes use of the Fourier series of nice periodic functions.

I'm also thinking that maybe one can also prove the isoperimetric inequality using calculus of variations (an infinitely many variable version of calculus). Any suggestions?

4. Can the problem be further generalized?

Sure, how about a higher dimensional version of the isoperimetric problem?

As it turns out, the isoperimetric problem is closely related to some other important inequalities in analysis, like the Sobolev inequalities (say for smooth functions with compact supports)

$\left(\int_{\mathbb{R}^n}|f(x)|^{\frac{n}{n-1}}dx\right)^{\frac{n-1}{n}} \leq C_n \int_{\mathbb{R}^n}|\nabla f(x)|dx$

(which allows one to control the size of a function from its derivatives).