Consider an octant $A \subset S^2$ on the sphere, for example the region $(\theta,\phi)\in[0,\pi/2]\times[0,\pi/2]$ in spherical coordinates. What is the subset $B \subseteq S^2$ with smallest spherical measure $\sigma^2(B)$, within which $A$ can be continuously rotated through $\text{360}^{\circ}$?
The smallest region I have found so far is defined by the parametric equations:
$$\begin{align} \gamma_1(t)&=\left(\frac{1}{\sqrt2},t,\sqrt{\frac{1}{2}-t^2}\right),\\ \gamma_2(t)&=\left(-\frac{1/2+t}{1+t},\frac{1}{\sqrt2},\frac{1}{\sqrt2}\frac{\sqrt{1/2-t^2}}{1+t}\right),\\ \gamma_3(t)&=\left(-\frac{1}{\sqrt2}\frac{\sqrt{1/2-t^2}}{1+t},-\sqrt{\frac{1}{2}-t^2},\frac{1/2+t+t^2}{1+t}\right). \end{align}$$
and their $\pi/2$ rotations about the $z$ axis. The region has measure $\frac{1}{\sqrt2}\left(\frac{1}{2}-\frac{\ln2}{\pi}\right)$ with respect to the normalised spherical measure ($\sigma^2(S^2)=1$).
This region and an octant rotating within it are depicted here projected onto the $xy$-plane. An interactive version on Desmos is depicted here.
I am interested in any improvements to this upper bound or methods to approach this question.
