On the Helly number for hyperplane transversals to unit balls

We prove two results about the Hadwiger problem of finding the Helly number for line transversals of disjoint unit disks in the plane, and about its h igher-dimensional generalization to hyperplane transversals of unit balls i n d-dimensional Euclidean space. These consist of (a) a proof of the fact t hat the Helly number remains 5 even for arbitrarily large sets of disjoint unit disks-thus correcting a 40-year-old error; and (b) a lower bound of d + 3 on the Helly number for hyperplane transversals to suitably separated f amilies of unit balls in R-d.