Let \(\mathbb{S}^d\) be the \(d\)-sphere . Bounds for the covering multiplicity for the optimal covering with spherical caps of constant radius on \(\mathbb{S}^d\) are found by Tóth [1] for \(d=2\), and by Böröczky and Wintsche [2, Theorem 1.1] for \(d\ge 3\). In the latter case, there exists a cover of balls with radius \(\varphi\in (0,\frac{\pi}{2})\) such that the covering multiplicity can be estimated by \(400d\ln d\).

References Link to heading

1. L. Fejes Tóth, Regular figures. The Macmillan Company, New York, 1964.
2. K. Böröczky and G. Wintsche, Covering the Sphere by Equal Spherical Balls, in Discrete and Computational Geometry, B. Aronov, S. Basu, J. Pach, and M. Sharir, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 2003, vol. 25, pp. 235–251.doi:10.1007/978-3-642-55566-4_10