ar X iv : m at h / 02 01 07 7 v 5 [ m at h . G N ] 1 5 Fe b 20 07 A family of pseudo metrics on B 3 and its application

Let B be the closed unit ball in R and S its boundary. We define a family of pseudo metrics on B. As an application, we prove that for any countable-to-one function f : S → [0, a], the set NMnf = {x ∈ S 2 | there exists y ∈ S2 such that f(x)−f(y) > ndE(x, y)} is uncountable for all n ∈ N, where dE is the Euclidean metric on R . 2000 Mathematics Subject Classification ; 57N05, 57M40 1 The family of pseudo metrics In this section we construct the family of pseudo metrics on the closed unit ball B ⊂ R. As usual, a nonnegative function d : B ×B3 → R is called a pseudo metric if 1. d(x, x) = 0 for all x ∈ B 2. d(x, y) = d(y, x) for all x, y ∈ B 3. d(x, y) + d(y, z) ≥ d(x, z) for all x, y, z ∈ B. Let S r ⊂ R be the 2-sphere with center O = (0, 0, 0) and radius 0 < r ≤ 1. We write dE to denote the Euclidean metric on R . A metric d on the set S r is called locally Euclidean if for all P ∈ S r , there exists t > 0 such that d(Q,R) = dE(Q,R) for all Q,R ∈ Bt(P ) = {S ∈ S r | d(P, S) < t}. Suppose that 0 < s ≤ 1. Let −P denote the antipodal point of P ∈ S r . Let α = sin (√ 2− s2 − s 2 ) , where 0 ≤ α < π/4.