Let $\Omega\subset\mathbb R^d$ be compact and convex, and denote by $\ell$ the normalised Lebesgue measure such that $\ell(\Omega)=1$. Let $N$ be an arbitrary but fixed integer.

In this post we set $d=2$ (we may consider general $d$ but here we focus on $d=2$). $V_1,\ldots, V_N\subset\Omega$ is said to be a $N-$partition of $\Omega$, if $\cup_n V_n = \Omega$, $\ell(V_n)=1/N$ and

$$V_m\cap V_n \mbox{ is either empty or a segment}.$$

If $\Omega=[0,1]^2$ or $\Omega=B(0,1):=\{x\in\mathbb R^2: |x|\le 1\}$, can we list all $N-$partitions for $N=3$ and $N=4$?

PS : $N=2$ can be treated by straightforward coordinate computation, while for $N\ge 3$ I don't have good intuition on geometry.