Let $S$ be a smooth, closed surface in $\mathbb{R}^3$,
and $\gamma$ a geodesic segment on $S$, i.e., a finite-length piece
of a geodesic.
Define $\gamma(w)$ as all the points of $S$ within
a distance $w$ of $\gamma$:
all the $x \in S$ such that the shortest distance on $S$ from $x$ to $\gamma$ is at most $w$.
One can imagine $\gamma(w)$ representing the path of a paintbrush of
width $2w$ as it traces $\gamma$:

_{(Image based on one at rdrop.com.)}

Q1. For a given $S$, which $\gamma(w)$ have the properties that (a) all of $S$ is covered by $\gamma(w)$, and (b) thearea product$\;|\gamma| \cdot w$ is minimized, where $|\gamma|$ is the length of $\gamma$.

In some sense, this is an optimal paint path: a paintbrush tracing $\gamma$ and spreading paint $\pm w$ covers the surface most efficiently. For example, for $S$ a unit-radius sphere, it seems that $\gamma$ a half-great circle, $|\gamma| = \pi$ is optimal with $w=\pi/2$, area product $\frac{1}{2} \pi^2$.

Q2. Is the semi-great circle indeed optimal for the sphere?

Q3. Are there other clear examples of optimal paintbrush geodesics?

Q4. Has this notion been studied before, perhaps in another guise?

Thanks for ideas/pointers!