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

Question

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!

Answer 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 $\frac {203}{\sqrt {51}}\approx 28.42$