RECTIFIABILITY OF MEASURES WITH LOCALLY UNIFORM CUBE DENSITY

• Published in: Proceedings of the London Mathematical Society 2003-01, Vol.86 (1), p.153-249
• Main Author: LORENT, ANDREW
• Format: Article
• Language: English
• Subjects: densities; rectifiability; sup norm
• ISSN: 0024-6115; 1460-244X
• DOI: 10.1112/S0024611502013710

The conjecture that Radon measures in Euclidean space with positive finite density are rectifiable was a central problem in Geometric Measure Theory for fifty years. This conjecture was positively resolved by Preiss in 1986, using methods entirely dependent on the symmetry of the Euclidean unit ball. Since then, due to reasons of isometric immersion of metric spaces into $l_{\infty}$ and the uncommon nature of the sup norm even in finite dimensions, a popular model problem for generalising this result to non-Euclidean spaces has been the study of 2-uniform measures in $l^{3}_{\infty}$. The rectifiability or otherwise of these measures has been a well-known question. In this paper the stronger result that locally 2-uniform measures in $l^{3}_{\infty}$ are rectifiable is proved. This is the first result that proves rectifiability, from an initial condition about densities, for general Radon measures of dimension greater than 1 outside Euclidean space. 2000 Mathematical Subject Classification: 28A75.