That second thing is perhaps your best example so far. The cube example suffers from construction. It's simple to see that the bounding spheres are fixed to th

That second thing is perhaps your best example so far. The cube example suffers from construction. It's simple to see that the bounding spheres are fixed to the cube's corners, which escape with dimension. While very interesting, this doesn't do much to constrain the ball's shape. Volume is also bad measure to use here, since it's already agreed that the volume of the entire sphere rapidly decreases. This is mostly due to the immense expansion of the reference cube. The ratio of the cap to the sphere is better, but you left that as an exercise to the reader, ignoring things like what height relative volume starts materializing in (a reasonable person might expect cos π/4). But picking a random direction from the surface is illuminating. It's hard to get rid of that effect without switching to spherical coordinates.