Does the perimeter of a polygon necessarily decrease if more edges are added to it, with the constraint of constant area?

A circle has the lowest perimeter for a 2D shape of a given area. To my understanding, it can also be approximated by a polygon of infinite sides. So, if I take an n-sided polygon and gradually add edges to it, keeping my area constant, will the perimeter also gradually decrease(Since I am approaching a circle)?

Thanks!

Solution 1:

Convexity is not sufficient. Take a unit square with area $1$ and perimeter $4$. Replace one side with an isosceles triangle with legs of length $100$. The area is now about $51$ and the perimeter is $203$. Scaling down linearly by $\sqrt {51}$ to make unit area leaves the perimeter $\frac {203}{\sqrt {51}}\approx 28.42$

Solution 2:

Depends on the details of how you add edges. You could get polygons looking like this.

Solution 3:

I guess that you are supposing that you are dealing with convex regular polygons. If you don't make explicit this hypothesis, the claim is false. In fact, by adding sides and keeping constant area you can obtain polygons with perimeters tending to infinity. You simply need to make the shape flatter and flatter. On the other side, if you consider a sequence of convex regular polygons with the same area and an increasing number of sides, the claim is true. Of course it need to be proved carefully. You need to prove that for any convex $n$-regular polygon inscribed in a unit circle, whose perimeter is $P_n$ and area $A_n$, the ratio $A_n/P_n^2$ is increasing with $n$.