Let $\gamma$ be a smooth curve in $\mathbb{R}^{3}$. We say that $\gamma$ is convex if it lies on the boundary of its convex hull. Recall that the convex hull of $\gamma$ is the set of all convex combinations of points of $\gamma$—or, equivalently, the minimal convex set containing $\gamma$.
I am a bit confused about this definition and its relation to the classical definition of convexity for planar curves.
In particular, I would like to ask the following question: if $\gamma$ is a spherical curve, i.e., it lies on $\mathbb{S}^{2} \subset \mathbb{R}^{3}$, is it true that $\gamma$ is automatically convex?