Covering the Sphere by Equal Spherical Balls

We show that for any acute ϕ, there exists a covering of Sd by spherical balls of radius ϕ such that no point is covered more than 400d ln d times. It follows that the density is of order at most d ln d, and even at most d ln ln d if the number of balls is polynomial in d. If the number of equal spherical balls is d + 3 then we determine the optimal arrangement.At the end, we described how our and other peoples results yield estimates for the largest origin centred Euclidean ball contained in the convex hull of N points chosen from the sphere.