  We show that the maximum number of unit distances or of diameters in a set of
n points in d-dimensional Euclidean space is attained only by specific types of
Lenz constructions, for all d >= 4 and n sufficiently large, depending on d. As
a corollary we determine the exact maximum number of unit distances for all
even d >= 6, and the exact maximum number of diameters for all d >= 4, for all
$n$ sufficiently large, depending on d.
