## Monday, March 5, 2012

### A Little Bit about Isoperimetric Inequalities

So, in Crelle XVIII, Jakob Steiner published the earliest known solution of the ordinary isoperimetric problem --- that is, among measurable plane figures with fixed perimeter, find which has the greatest area. I'm sure something like this result had been known considerably earlier; Euler and later Lagrange had formulated and solved rather general extremum problems for curves, and the isoperimetric example isn't difficult to solve. Variational problems were posed by one of the Bernoulis --- most notably the brachistochrone, of which several solutions were published simultaneously (Oh! For the good old days!). My source for Steiner's priority also claims he's the first person to have formulated the problem (or at least to have published such a formulation). So I'm inclined to suspect that Newton and Leibniz and them all thought the thing so painfully obvious that they never mentioned it.

The isoperimetric problem is considerably simpler than the brachistochrone. In fact, there are a number of similar problems which have strikingly similar solutions; for instance, among rectangles having fixed perimeter, that with the greatest area is a square, which is practically a proposition in Euclid, and quickly generalizes to the elsewhere-famous Cauchy-Shwartz-Буняковский inequality. If we said "triangles" instead of "rectangles", we'd get an equliateral triangle as the solution, and if you want to keep guessing at the best answers, you'll see this post is actually a love-song to symmetry.

The isoperimetric problem itself has rotational symmetry. Since rotating a plane figure changes neither its perimeter nor its area, it should be clear that if we had any extremal isoperimetic figure, then we could get more extremal figures by rotating the first one. That is, we could get more if there were more --- if our extremal isoperimetric figure didn't have the same symmetry as the isoperimetric problem itself. But there aren't more, there's only one. And there's a good reason there's only one, but we'll get to that another time.

For now, pleasant dreams!

Belfry Bat said...

... on poking around a little further, it seems there is also symmetry lurking in one of the Bernouli's solutions to the brachistochrone... maybe that'll come later, too!

Salome Ellen said...

I'm pleased that after a lapse of the almost 40 years since college -- and I don't think I heard the term isoperimetric back then -- my first to your first paragraph was "Circle, of course!"

Salome Ellen said...

That's "first REACTION". I couldn't type back then either....

Belfry Bat said...

Yes! Anyone who has ever blown soap bubbles can figure it out; we mathematicians are a weird lot in that we want to figure all this stuff syllogisticaly. And Steiner actually did, in this case.

(btw, If my typing/spelling ever improves, I'll spare you the bragging.)

Enbrethiliel said...

+JMJ+

One of my high school Algebra teachers insisted that we all keep "math journals." I think she'd really love your posts.

Belfry Bat said...

Hmmm... On the one hand, I like appealing to enthusiastic teachers of Algebra (and also teachers of Problem Solving by Symbolic Reasoning, though that takes longer to say). On the other hand, I have memories that could comfortably be vaguer about being chronically unable to fulfill others' expectations vis-a-vis journal-writing. Ah, well.

I hope I may take it that you also like these tidbits? ;-)