On the optimality of random partial sphere coverings in high dimensions

Given $N$ geodesic caps on the unit sphere in $\mathbb{R}^d$, whose total normalized surface area is one, what is the maximal proportion of the sphere that their union can cover? In this work, we provide an asymptotically sharp upper bound for an antipodal partial covering of the sphere by $N=N(d)$ congruent caps in the regime $N(d)\to\infty$ and $\ln N(d)=o(\sqrt d)$, showing that the maximum proportion covered approaches $1 - e^{-1}$ as $d\to\infty$. We discuss the relation of this result to the optimality of random polytopes in high dimensions, the limitations of our technique via the Gaussian surface area bounds of K. Ball and F. Nazarov, and its applications in computer science theory.