[hxypjqcg] slicing off vertices of a polytope

start with any convex polyhedron and its inscribed sphere.  for each vertex, project it onto the sphere, construct a tangent plane at that point on the sphere, then slice off the portion of the polyhedron above the tangent plane.

this operation can be repeated.  if we started with a symmetric polyhedron, after each round of slices, each new polyhedron keeps that symmetry.  consider coloring the faces in a pretty way.  infinite repetition converges to a sphere.

this can be done in any number of dimensions, including 2.

start with a d-dimensional hypercube.  by the curse of dimensionality, its volume is much larger than its inscribed hypersphere.  does the first round of slices, chopping off 2^d corners, reduce the remaining volume to approximately that of the hypersphere?  if not, how many rounds does it take?

in 3D, the first round of slices cuts off more than the cuboctahedron, leaving a truncated octahedron but not the Archimedean solid: all edges are not all the same length.  both the cuboctahedron and Archimedean truncated octahedron have conveniently simple coordinates, but this truncated octahedron with all faces tangent to a sphere does not.