This question is motivated by this question.
Description
Let us to start with any convex figure. without loss of generality we may assume that length of its perimeter is equal to 1. Then let's execute the following algorithm:
- Choose line $L$ which goes through point $p_0$ on $\partial \Phi$ and divides $\Phi$ into two subfigures $\Phi'$ and $\Phi''$ such that $|\partial \Phi'| = |\partial \Phi''|$.
If $S(Φ′)\leq S(Φ″)$ then substitute $∂Φ″$ with the reflection of $∂Φ′$ around the $L$.
Else vice-versa. We get other figure $\chi$. Assign $\Phi = \chi$ Substitute $p_0$ with other point $p_0'$ which away from $p_0$ by $1/\sqrt{2}$ along $\partial \Phi$. Assign $p_0 = p_0'$
Return to step 1.
Starting with a square @Aretino obtained this:
So what property is related to this? Something like find convex closed curve possessed given
length with the smallest area does not work, obviously.
Not every starting figures leads to this jagged circle. For example, rectangle does not.