  We begin the study of the notion of diameter of an ideal I of a polynomial
ring S over a field, an invariant measuring the distance between the minimal
primes of I. We provide large classes of Hirsch ideals, i.e. ideals with
diameter not larger than the codimension, such as: quadratic radical ideals of
codimension at most 4 and such that S/I is Gorenstein, or ideals admitting a
square-free complete intersection initial ideal.
