Lower bounds for the Lipschitz constants of some classical fixed point free maps

We find lower bounds for the set of Lipschitz constants of a given Lipschitzian map, defined on the closed unit ball of a Hilbert space, with respect to any renorming. We introduce a class of maps, defined in the closed unit ball of ℓ2, which contains the classical fixed point free maps due to Goebel-Kirk-Thelle, Baillon, and P.K. Lin. We show that for any map of this class its uniform Lipschitz constant with respect to any renorming of ℓ2 is never strictly less than π2.