The geometry of uniform measures

Abstract

Uniform measures have played a fundamental role in geometric measure theory since they naturally appear as tangent objects. They were first studied in the groundbreaking work of Preiss where he proved that a Radon measure is n-rectifiable if and only if the n-density at almost every point of its support is positive and finite. However, very little is understood about them: for instance the only known n-uniform measures not supported on an affine n-plane were constructed by Preiss in 1987. In this thesis, we prove that the Hausdorff dimension of the singular set of any $n$-uniform measure is at most n-3. Then we characterize 3-uniform measures with dilation invariant support and construct an infinite family of 3-uniform measures all distinct and non-isometric, one of which is the Preiss cone.