# Derivation of isoperimetric inequality from ideal gas equation

As I was going through my notes on statistical physics it occurred to me that from a classical perspective, I could derive the isoperimetric inequality from the ideal gas equation and its associated Maxwell distribution. If you’re unfamiliar with the isoperimetric inequality it basically states that for a given Jordan curve $C$ with fixed perimeter, the circle maximises the internal area. So my assertion is basically equivalent to saying 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. After showing that the volume must be bounded as a result of the Whitney-Loomis inequality, 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 $M$,

$P^*$ everywhere on $\partial M$ implies $M$ is spherical

So far I have a semi-rigorous mathematical argument which can be summarised as follows. 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.

A more complete account of what I’ve attempted so far can be found here in my essays section. Meanwhile, despite numerous google searches, it appears that this result hasn’t been rigorously proven by mathematical physicists so far. In fact, I asked the question on the mathoverflow and from the responses it’s clear that the result isn’t obvious. It might require very careful averaging arguments.

In fact, it might interest the reader to know that the isoperimetric inequality itself eluded many of the greatest mathematicians in history including Euler until the efforts of Jakob Steiner in the 19th century finally led to its proof.