The Hausdorff Measure of Sets Which Link in Euclidean Space

Abstract

This chapter discusses an elementary lower bound for the Hausdorff p-dimensional measure of a compact set A ⊂ Rn, which links a compact (n – p – 1)-dimensional set B ⊂ Rn ∼A. For each n ≥ 2, there exists a positive constant c, such that ℋ p (A) ≥ c dist (A, B)p, whenever p is an integer in [1, n –1] and B is a topological (n – p – l)-sphere, which is not contractible in Rn ∼ A. Inequality is a consequence of a result, which states that each compact set A ⊂ Rn with ℋ p (A) = rp , r ∈(0, ∞), can be covered by a compact set C ⊂ Rn, such that Rn ∼ C is (n – p – l)-connected and such that each point of C Lies with distance br of A, where b is a positive constant which depends only on n. The chapter also presents an application of inequality to n-dimensional quasiconformal mappings.