Bouligand dimension and almost Lipschitz embeddings

Eric Olson

Vol. 202 (2002), No. 2, 459–474

DOI: 10.2140/pjm.2002.202.459

Abstract

In this paper we present some new properties of the metric dimension defined by Bouligand in 1928 and prove the following new projection theorem: Let dim_b(𝒜−𝒜) denote the Bouligand dimension of the set 𝒜−𝒜 of differences between elements of 𝒜. Given any compact set 𝒜⊆ R^N such that dim_b(𝒜−𝒜) < m, then almost every orthogonal projection P of 𝒜 of rank m is injective on 𝒜 and P|_𝒜 has Lipschitz continuous inverse except for a logarithmic correction term.

Milestones

Received: 5 April 2000

Revised: 3 November 2000

Published: 1 February 2002