Suppose you would like to optimally place congruent spherical caps on the unit sphere in , each cap having volume . (Here volume refers to the normalized surface area; the sphere has volume ). How much volume of the sphere can you cover?

Clearly the whole sphere cannot be covered: the caps must overlap if (which we assume). Moreover, it is not immediately obvious that a substantial proportion of the sphere can be covered this way. In fact, if we *do* insist that the caps (say, again, of volume ) are disjoint, the proportion that can be covered is at most (Kabatianskii and Levenshtein, 1978, MR0514023).

This where the *probabilistic method *shows its effectiveness: placing the caps at random leaves an expected uncovered volume of . Thus one can cover a proportion of the sphere.

Can one do better? Can we hope to cover a proportion ? Or even for some ? The question was asked by Gil Kur, a reader of our book, and we have no idea about the answer! There is nothing special about , you may replace it by other values.

There are few lower bounds on the density of spherical coverings (incidentally, the terms “overlap” or “redundancy” would be more logical). The general upper bound due to Rogers via random covering (Corollary 5.5 in our book) achieves a density of order . A lower bound of order holds for caps whose radii–in the geodesic metric–are either very close to (since caps are needed) or to (via the *simplex bound* of Coxeter-Few-Rogers), but little seems to be documented in the literature for the intermediate range. A conjectural spherical simplex bound would imply a lower bound of for all radii, see Conjecture 6.7.3 and the open Problem at the end of Section 6.8 in Károly Böröczky’s book Finite Packing and Covering.