A family of pseudo metrics on B^3 and its application

Abstract

Let B^3 be the closed unit ball in R^3 and S^2 its boundary. We define a family of pseudo metrics on B^3. As an application, We prove that for any countable-to-one function f:S^2\to [0,a], the set NM^n_f={x\in S^2 | there exists y\in S^2 such that f(x)-f(y)>nd_E(x,y)} is uncountable for all natural number n, where d_E is the Euclidean metric on R^3.