For $n\geq 4$, let $V_n$ be the maximum volume of the convex hull of $n$ points on the unit sphere (in $\mathbb{R}^3$, although information on higher dimensions is welcome as well). I'm sure the problem of computing $V_n$ has been extensively studied and has a standard name: what is this name?

For which values of $n$ is the exact value of $V_n$, and/or a configuration realizing it, known (at least combinatorially)? And, for small $n$, what are the best known configurations even if they are not proved to be optimal? Essentially, I'm looking for a list analogous to what Wikipedia lists for the Thomson problem but for the volume of the convex hull instead of electrostatic potential energy.