Matt discusses haircuts for guys with a little bit less on top
[ This is the simplified version without explicit time-dependent analysis. I was thinking of publishing this but not sure what journal would take it. Maybe I could submit it to GQ ? ]
The optimal haircut for a guy pretty much has p(x)=constant, where x is the spatial position, and p is the hair length at that positioin. For now let us just consider x as a linear variable although it can be easily generalized to a vector.
Let D(x) be the hair-follicle density. Ideally, D(x)=constant.
But some guys have a hair follicle density D(x) that is non-uniform, which is a non-ideality.
So if we set p(x) proportional to D(x), then p(x) will be non-uniform which is not optimal.
There are a couple solutions.
First, you can set p(x)=0. That’s the Michael Jordan look. It is certainly a solution although it is unstable as a function of time, so that it requires constant haircuttery.
Another solution is to set p(x)=1/D(x), which may be denoted as the inverse haircut. This is favored by many hairdressers you will find at SuperCuts, etc. Avoid such hairdressers. The problem here is again the time dependence. The length is a function of time, and what you will find with an inverse haircut is that it looks good at t=0, but it fails with increasing time.
The last solution is to have a non-local solution for p(x), also known as the comb-over. This solution becomes increasingly untenable as D(x) has more shape to it. The observer can easily see the non-local solution.
Currently, our research group is working on advanced mathematical models and simulations to further analyze and address such problems !