Given a circle, we want to divide it into $n$ connected equally sized pieces. In such a way that the total length of the cutting is minimal. What can we say about the solution for each $n$. Are they unique (up to some symmetry). Do all cuttings arise from the intersection of three straight segments with angle 120º?

These are conjectured to be the best solutions for $n \in \{2, 3, 4, 7\}$

Case $n= 2$ goes as follows: For simplicity the radius is $1$. If there is only one or no points on the circumference, then we know the minimal curve is the circumference, which has length $\frac{2\pi}{\sqrt{2}}$ bigger than 2 and we are done. Hence, we have at least two points in the circumference. If they are in the same diameter, the shortest curve is the line, so the cutting has at least length 2.

Now, pick a diameter parallel to $\overline{AC}$, which leaves both points on the same side. Since the curve connecting both needs to have area $\pi/2$ in needs to contain at least one point on the other side of the diameter.

Now, $\overline{AB}$ is bounded below by a straight line, same with $\overline{BC}$. And the sum of those two is smaller than $\overline{AO} + \overline{OC}$ where $O$ is the center of the circle.

For $n\in\{3, 4, 7\}$ I don't have a proof and they might not be minimal.

As for the conditions on the cuttings, probably something like picewise $C^2$ is enough.

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