A family of pseudo metrics on B and its application

We define a family of pseudo metrics on B3 and study elementary properties of the associated metric spaces. As an application we prove that, for any a > 0 and for any countable-to-one function f from (S2, dE) to [0, a], the set NMf = {x ∈ S2 | ∃y ∈ S2 such that f(x) − f(y) > ndE(x, y)} is uncountable for all n ∈ N, where dE is the standard Euclidean metric on S 2 = {(x, y, z) ∈ R3 | x2 + y2 + z2 = 1}.