On uniform properties of doubling measures

Abstract

In this paper we prove that if (X, d, n) is a metric doubling space with segment property, then inf r(E)/r(B) > 0 if and only if inf μ.(E)/μ(B) > 0, where the infimum is taken over any collection C of balls E, B such that E ⊂ B ⊂ X. As a consequence we show that if A is a linear metric doubling space, then it must be finite dimensional. ©2001 American Mathematical Society.