Starting with a convex polyhedron $P_1 \subset \mathbb{R}^3$, replace that with $P_2$, the convex hull of the midpoints of the edges of $P_1$. Continuing this process, we obtain a series of polyhedra approaching a smooth body $B$ (or at least, I think it approaches a smooth body). See above for $P_1,\ldots,P_5$—not to the same scale.
Q1. Is $\lim_{n \to \infty} P_n$ $C^2$-smooth, starting with non-degenerate $P_1$? Or only $C^1$-smooth? Or only $C^0$?
Q2. Does $P_n$ approach an ellipsoid as $n \to \infty$, for every (non-degenerate) starting $P_1$? [My guess: No.]
These are, in some sense, less sophisticated versions of my earlier question, “Derived”polyhedra and polytopes, which focussed on face centroids rather than edge midpoints. But here I am asking specific questions on smoothness and the limit shapes. Still, perhaps @GjergjiZaimi's answer there holds.
The limit objects are almost subdivision surfaces, but not quite.
