On coverings of spheres by convex sets (Q801297)

Continuing his earlier investigations on coverings of the Euclidean space by congruent copies of given convex sets [Am. Math. Mon. 87, 286-287 (1980; see the preceding review), Mathematika 29, 18-31 (1982; Zbl 0496.52011), J. Comb. Theory, Ser. A 34, 71-79 (1983; Zbl 0509.52006)] the author proves the following result: Given a sequence of closed convex subsets of the n-dimensional unit sphere \(S^ n\) such that the sum of their n-dimensional measures exceeds a certain constant depending only on n, the convex sets can be rotated such as to cover \(S^ n\). For \(n=2\) and in case the geodesic diameters of the convex sets have small upper bounds, there are more precise results. These results are all closely connected with the problem of covering Euclidean space by congruent copies of given unbounded convex sets.