What if we put the biosphere on the outside?

Ideally we'd like Earthlike conditions on the outside surface; that is, one gee of gravity and a temperature of about 300 K. Can we get this with main sequence stars?

The mass-luminosity relation is [reference 1]

The gravitational acceleration g for a mass M at a distance R is:

Finally, the radiation power law relates the luminosity L to the area A and temperature T:

We can then combine these three equations into a single one relating M (the mass of our star), T (the temperature of our Dyson shell), and g (the gravitational acceleration at the surface of our sphere). First we start by equating L via the luminosity of the star and the radiation power law:

The area A of a sphere is simply 4 pi R², so we can write

We can use our gravitational acceleration equation to relate R² to M:

Assuming our sphere appoximates a blackbody (e ~= 1) and then substituting ideal conditions (g = 9.81 m/s², T = 300 K), we find that M must vary between 0.054 and 0.079 masses solar (the variance is caused by 3.5 <= nu <= 4.0).

By comparison, the "end of the main sequence" -- that is, the theoretical point at which a main sequence star is unable to sustain itself by hydrogen fusion -- is at about 0.08 masses solar, but is not precisely known. Thus it might be possible to have Dyson shells with Earthlike conditions on their outside surfaces around the smallest hydrogen-burning stars in the Universe.

One other advantage to using red dwarfs is their immense lifespan -- the least-massive red dwarfs can last hundreds of billions or trillions of years.