A local version of the Projection Theorem

Toby O'Neil

We prove the following theorem:

Suppose that $1\leq m\leq n$ are integers and $\mu $ is a Borel measure on $\Real^n$ are such that for $\mu$ almost every $x$ ,

The upper and lower $m$-densities of $\mu $ at $x$ are positive and finite.
If $\nu $ is a tangent measure of $\mu $ at $x$ then for all $V\in\G (n,m)$ the orthogonal projection of the support of $\nu $ onto $V$ is a convex set.

Then $\mu $ is $m$-rectifiable.

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Created:13 August 1996
Modified:15 August 2000