Redoing the napkin math because I can't find the original and I've somehow nerd-sniped [0] myself. Assume a sphere that contains a cubic meter of air and is neutrally buoyant:
Vacuum volume 1.00 m^3
Vacuum radius 0.62 m
Air density 1.23 kg/m^3
Displaced mass 1.23 kg
Diamond density 3250 kg/m^3
Diamond volume 3.77E-04 m^3
Sphere surface area 4.84E+00 m^2
(Thin shells can be approximated by ignoring curvature)
Shell thickness 7.79E-05 m
Shell thickness 77.94 μm (microns)
Human hair thickness 17 to 181 μm (microns)
So the big vacuum-ball somehow needs to withstand crushing using only the strength of a diamond skin, and that skin is around as thick as (some) human hairs. It has to be thinner if you want it to float with a payload.
Remember: The walls of party-balloons and soap-bubbles are a very different situation: They can have extremely thin skins because the gas inside is doing almost all the "don't get crushed" work, the skin is really just there to keep the two gases from mixing.
Perhaps they use a diamond frame filled in with graphene instead. Still seems pretty unlikely though. And probably too easy to puncture, ruining the vacuum.
Or perhaps they just build surfaces out of more complicated things than carbon. Hydrocarbons could be as easily assembled as diamond with similar techniques, and are a lot lighter than pure carbon. Plus they would want to put electronic advertising on the surface of the ball anyway, so it’s got to have a bit more going on to form LEDs and whatnot.
I was thinking about smaller bubbles and aerogels, but when the volume changes by a factor of 0.10, the thickness-budget only changes by a factor of ~0.46.
So while many smaller spheres--or pockets sharing walls--might be easier to engineer, the square-cube law [0] eats away at your lifting-power as everything becomes mostly-diamond with very little air-displacing vacuum.
Likely, but even a cubic meter is already a large step up from the scales in the book, ex:
> Each aerostat in the dog pod grid was a mirror-surfaced, aerodynamic teardrop just wide enough, at its widest part, to have contained a pingpong ball.
It doesn't say they aren't relying on some kind of down-thrust, but at that point I'd wager it's much easier to skip the whole marginal buoyancy part entirely. Use those conductive diamonds for an ionic thruster or something.
Remember: The walls of party-balloons and soap-bubbles are a very different situation: They can have extremely thin skins because the gas inside is doing almost all the "don't get crushed" work, the skin is really just there to keep the two gases from mixing.
[0] https://xkcd.com/356/