For A>0, the A-Lipschitz minorant of a function f is the greatest function m such that m > =f and |m(s)-m(t)| > A|s-t|. We investigate the A-Lipschitz minorant of a real-valued two-sided Lévy process X and in particular the set of points where X meets its minorant, known as the contact set. The contact set is stable and regenerative and hence can be thought of as the range of some...