Rectifiability of Measures with Locally Uniform Cube Density

The conjecture that Radon measures in Euclidean space with positivefinite density are rectifiable was a central problem in GeometricMeasure Theory for fifty years. This conjecture was positivelyresolved by Preiss in 1986, using methods entirely dependenton the symmetry of the Euclidean unit ball. Since then, dueto reasons of isometric immersion of metric spaces into l andthe uncommon nature of the sup norm even in finite dimensions,a popular model problem for generalising this result to non-Euclideanspaces has been the study of 2-uniform measures in . The rectifiability or otherwise of these measureshas been a well-known question. In this paper the stronger result that locally 2-uniform measuresin are rectifiable is proved. This is the first result that proves rectifiability, from aninitial condition about densities, for general Radon measuresof dimension greater than 1 outside Euclidean space. 2000 MathematicalSubject Classification: 28A75.