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![MidPoints123][1]
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![Midpoints34][2]
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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 $P_1,\ldots,P_5$&mdash;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](http://mathoverflow.net/q/149770/6094),
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](http://mathoverflow.net/a/149855/6094) holds.


The limit objects are almost [subdivision surfaces](http://en.wikipedia.org/wiki/Subdivision_surface), but not quite.


  [1]: https://i.sstatic.net/aYDhQ.jpg
  [2]: https://i.sstatic.net/LRhdk.jpg