This question concerns a process that iterates intersection of randomly rotated planar shapes.

Start with a simply connected region $R_0$ in the plane, and let $c_0$ be the centroid of $R_0$. Rotate $R_0$ about $c_0$ a random angle; call the result $R'_0$. Set $R_1 = R_0 \cap R'_0$. And repeat, always rotating about $c_0$, and computing $R_{i+1} = R_i \cap R'_i$. It is not difficult to see that, if $c_0 \in R_0$, then the process converges to a disk:

^{ Rotation center $c_0$ fixed to centroid of $R_0$: $\rightarrow$ disk. (Scale changes frame-to-frame.) }

If $c_0 \not\in R_0$, then eventually the empty set is reached.

My question concerns the process where the rotation center moves each step to $c_i$, the centroid of region $R_i$. Then sometimes, even when $c_0 \not\in R_0$, the process converges to a disk:

^{ Rotation center $c_i=$ the centroid of $R_i$: $\rightarrow$ disk. (Scale changes frame-to-frame.) }

And sometimes, for the same shape, it leads to the empty set:

^{ Rotation center $c_i=$ the centroid of $R_i$: $\rightarrow \varnothing$ (in the 5th step not shown).}

^{ (Scale changes frame-to-frame.) }

For the process that moves the rotation center $c_i$ to the centroid of $R_i$ at each step:

. What characteristics do the shapes $R_0$ possess that lead to a disk with high probability? And what characteristics lead to $\varnothing$ with high probability?Q

For example, I believe that if $R_0$ is convex, then the process always leads to a disk (not generally the same disk as when the center is fixed at $c_0$ throughout). But I am having difficulty seeing any regularity for nonconvex $R_0$.