On the Helly Number for Hyperplane Transversals to Unit Balls

We prove some results about the Hadwiger problem of nding the Helly number for line transversals of disjoint unit disks in the plane, and about its higher-dimensional generalization to hyperplane transversals of unit balls in d-dimensional Euclidean space. These include (a) a proof of the fact that the Helly number remains 5 even for arbitrarily large sets of disjoint unit disks|thus correcting a 40-year-old error; (b) a lower bound of d+3 on the Helly number for hyperplane transver-sals to suitably separated families of unit balls in R d ; and (c) a new proof of Danzer's theorem that the Helly number for unit disks in the plane is 5.