Just a bit more "data" in the form of images. I started with a "random polyhedron"
built from the convex hull of a small number of points. Then I iterated the process
of replacing each face by its centroid, i.e., $\alpha=\beta=\gamma=\frac{1}{3}$
in my original description, and
again taking the convex hull. At least visually, the data support Gjergji's suggestion
that the process converges to ellipsoids (of different axes lengths).
Joseph O'Rourke
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