The following is just a rough version of the question, which is made more clear below:
How can it be proven/disproven that the fastest route for a racecar making a quarter-circle turn is the route given by a section of a bigger circle that intersects the outer part of the track at the start and end of the turn and intersects the inner part of the track halfway through the turn?
In the figure below, I have drawn a circle of radius $R$ so that it intersects 1) a circle with radius $r_2$ at points that are $\frac{\pi}{2}$ radians apart in the circle defined by $r_2$, and 2) a smaller circle of radius $r_1$ inscribed in the one with radius $r_2$. The figure should make it clear.
I have calculated $R$ to be $$R=\frac{r_1^2+r_2^2-\sqrt{2}r_1r_2}{2r_1-\sqrt{2}r_2}$$ which makes sense since $R\rightarrow \infty$ as $r_1 \rightarrow \frac{\sqrt{2}}{2}r_2$, in which case you can draw a straight line.
Now, the length of the arc drawn out by the bigger circle from one point of intersection to the other is given as $\theta R$, where $$\theta = 2\arctan\left(\left(2\left(\frac{R}{r_2}\right)^2-1\right)^{-1/2}\right)$$
Now returning to the question: We assume that the speed at which the racecar is going at any point in the turn is proportional to $\sqrt{r}$, where $r$ is the radius of turning at that point (i.e. the curvature at that point is $1/r$).
For a track with constant width $r_2-r_1$ (with $\frac{\sqrt{2}}{2}r_2<r_1<r_2$), is the route through a quarter-circle turn that minimizes the time spent in said turn given by a section of a circle with length $\theta R$ (as seen in the figure)?
The reason I think it is given by $\theta R$ is based purely on intuition.
I guess the problem will most easily be solved using calculus of variations, as it is somewhat similar to the Brachistochrone problem (with the added constraint of the width of the track), but I don't know how to attack it.
Thanks!
EDIT 1: This question addresses the more general question of finding the optimal route around a track, but has an algorithmic approach instead. If someone coincidentally has a script with such an $A*$ algorithm at hand, they could test it on a quarter-circle turn and see whether or not the route of least curvature follows the arc of a bigger circle?
EDIT 2: As Jens has suggested in the comments, some kind of speed limit should be introduced. I think it is sufficient to demand that the function given by the speed of the car should be differentiable (after all, infinite acceleration isn't very realistic).