It is often assumed that, given $n$, the regular $n$-gon will make the most efficient use of perimeter for area. I have never seen this proven. Anyone have something slick?
(That is, how can we prove that, given some number of sides, the regular polygons maximize area for a fixed perimeter? Or minimize perimeter for a fixed area, or maximize the ratio, $\frac{A}{p}$, or minimize the reciprocal… You get the point.)
This is easy enough to show for rectangles, and I've done it for triangles as well. I'm struggling find a generalizable method, since the space of $n$-gons with fixed perimeter (up to congruence) is $2n-4$-dimensional, I believe. (Another problem I'm thinking on.)
Any thoughts on a general proof of this claim?
Best Answer
There must be a simple proof that doesn't involve analysis, but I'm too lazy to think of that now. If we allow analytic arguments, there is a simple plan. It is kind of rough.
This plan is kind of sketchy, but it looks perfectly doable. The main advantage is that you don't have to invent anything smart about the polygon as a whole, you just need to look at 3 or 4 consecutive vertices and come up with ways to increase the area if the polygon isn't regular.