In the following I consider spherical distances.
Say a subset $X$ of the sphere $\mathbb{S}^n$ is convex if it contains all geodesic segments between its points. Let $k<\frac{2\pi}{3}$, and suppose that $X$ has diameter $<k$. What is the optimal function $f(k)$ such that $X$ has to be contained in a disk of radius $f(k)$?
(For $k\geq\frac{2\pi}{3}$ we have $f(k)=\pi$, considering an equilateral triangle)
Sharp upper bounds for $f$ could also be useful. In case this is heavily dependent on the dimension $n$, I am happy with $n=3$.