Given \( N \) geodesic caps on the unit sphere in \( \mathbb{R}^d \), and whose total normalized surface area sums to one, what is the maximal surface area their union can cover? In this work, we provide an asymptotically sharp upper bound for an antipodal partial covering of the sphere by $N \in (ω(1),e^{o(\sqrt{d})})$ congruent caps, 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.

翻译：给定单位球面 \(\mathbb{R}^d\) 上的 \( N \) 个测地帽，其归一化表面积之和为一，这些测地帽的并集所能覆盖的最大表面积是多少？本文针对由 \( N \in (ω(1),e^{o(\sqrt{d})}) \) 个全等测地帽构成的对映部分覆盖，给出了一个渐近尖锐的上界，证明当 $d\to\infty$ 时，最大覆盖比例趋近于 \( 1 - e^{-1} \)。我们讨论了该结果与高维随机多胞形最优性的关联，通过 K. Ball 与 F. Nazarov 的高斯表面积界分析了我们方法的局限性，并探讨了其在计算机科学理论中的应用。