TITLE: What's the biggest circle that can fit between 2 Gaussian curves?
QUESTION [17 upvotes]: What's the biggest radius possible for a circle to fit completely between the curve $y = e^{-x^2}$ and $y = -e^{-x^2}$ ?
This isn't homework, I was just thinking about this randomly. I know calculus will be involved to find the radius, but I just can't figure out how to proceed. r=1 is too big

REPLY [29 votes]: The circle by the points $(\pm x,\pm e^{-x^2})$ and centered at the origin has the squared radius
$$r^2=x^2+e^{-2x^2},$$
which, by differentiation, achieves the minimum
$$r^2_{min}=\frac{\ln 2+1}2.$$


Addendum:
Let us consider the offset curve at distance $2r_{min}$ (as if the circle was rolling on it). From the parametric equations
$$\begin{cases}x=t,\\y=e^{-t^2},\end{cases}$$
we draw the green curve below
$$\begin{cases}x'=x-\frac{2r_{min}2te^{-t^2}}{\sqrt{1+4t^2e^{-2t^2}}},\\
y'=y-\frac{2r_{min}}{\sqrt{1+4t^2e^{-2t^2}}}.\end{cases}$$

The portion of the curve between the two cusps$^*$ mustn't be considered, because it corresponds to a section where the radius of curvature is smaller than $r_{min}$ so that the circle is externally tangent.
This shows that no larger circle can be tangent to the upper curve without passing the lower one and validates the intuitive choice of the center at the origin.
$^*$ The cusps are at the same time the tangency points of the circle to the lower curve.