# rigorous derivation of isoperimetric inequality from ideal gas equation

I'm an undergraduate math student that learned about classical ideal gases and the associated maxwell-boltzmann distribution for particle velocities in a statistical physics course. Now, starting from this knowledge and assuming that there are no collisions between particles, I tried to show that an ideal gas maximises the enclosed volume.

More formally for any compact Euclidean manifold $M$ in $\mathbb{R}^3$ with elastic boundary $\partial M$ containing a macroscopic number of gas particles, a gradual increase in temperature ultimately results in a sphere.

My method of proof is to take for granted that $\partial M$ allows a maximum pressure $P^*$ and that if we gradually increase the temperature $T$ of the particles inside $M$,

$$P^* \text{ everywhere on \partial M} \implies M \text{ is spherical} \tag{*}$$

I think this method of proof would work but I realised that it's not easy to fill in the details considering that I assume that this balloon $M$ is contained in a gas with a different temperature and pressure. For this reason, I'd like to know whether this result has already been proven rigorously for a macroscopic number of particles in order to compare my work.

I think this might be as hard as proving the double bubble conjecture so it might be that this result has not yet been rigorously demonstrated.

Note 1: I am interested in the special case of Hookean elasticity as I think that Hookean elasticity would be a sufficiently good approximation. I also assume that the boundary $\partial M$ has negligible thickness.

Note 2: So far I have the following handwavy argument that points to a proof. Assuming that $P^*$ has been attained uniformly on $\partial M$, the magnitude of the force on $\partial M$ must be approximately constant all over $\partial M$. In consequence, by the Maxwell-Boltzmann distribution and momentum conservation, the distance between opposite extremities of the balloon must be approximately constant.

For the $n$-dimensional phase space of $n$ particles on a line the isoperimetric inequality of an ideal gas was derived in: Phase space measure concentration for an ideal gas (2009).