Covering the unit sphere in $\mathbf{R}^n$ with $2n$ congruent disks

Question

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 $n\geq 4$.

Answer 1

For $n=3$ the answer is yes, as was shown by Fejes Tóth in 1943; see the Theorem on p.34 of his book Regular Figures. For $n=4$ the answer is also positive as shown in the 2000 paper, The blocking numbers of convex bodies, by Dalla, Larman, Mani-Levitska, and Zong. In higher dimensions, the problem appears to be open according to Conjecture 1.3 in the 2003 article, Covering the sphere by equal spherical balls, by Böröczky and Wintsche. In a 2015 paper, On the Distance of Polytopes with Few Vertices to the Euclidean Ball, Tikhomirov showed that $|v_i|\geq C\sqrt{n}$ where $C$ is an absolute constant. Another proof of this fact follows from a result of Nazarov included in the appendix to the 2022 paper Shortest closed curve to contain a sphere in its convex hull, which is discussed in this post. Here one may also estimate that $C\geq 0.27$.