Suppose we are given an integer $N\ge4$, and we have to pick $N$ points in a unit ball in $\mathbb R^3$ to maximize the volume of their convex hull. Are those points necessarily on the surface of the ball? Can we generalize the result to $\mathbb R^n$?
For the similar problem in $\mathbb R^2$ the answer seems obvious: we always have to pick the vertices of a regular $N$-gon inscribed in the disk, so they always lie on the disk boundary.
Take a point $Q$ in the interior of the convex hull of points $P_1,\cdots,P_N$ and extend the rays $QP_k$ to the points $P_k^\prime$ on the surface of the ball to get a convex hull containing the old yet with volume not less than the old.