Let $v_i$ be $2n$ points in $\mathbf{R}^n$, with equal distance $|v_i|$ from the origin. Suppose that the convex hull of these points contains the unit ball. Is it known that $|v_i|\geq\sqrt{n}$? Presumably, the optimal configuration is given by the vertices of a cross polytope, i.e., when $v_i$ are placed on the coordinate axis with $|v_i|=\sqrt{n}$.

The problem may also be phrased in terms of the smallest radius for $2n$ congruent spherical disks to cover the unit sphere $\mathbf{S}^{n-1}$ (given by points $x\in\mathbf{S}^{n-1}$ with $\langle x, v_i\rangle\geq 1$). So I imagine that either the answer is known, or else this is a well-known problem in sphere covering. References would be appreciated. The case $n=2$ is obvious, and $n=3$ seems doable. But I am not sure about $n\geq 4$.