A local version of the Projection Theorem
Toby O'Neil
Appeared in The Proceedings of the LMS, July 1996.
Abstract
We prove the following theorem:
Suppose that \(1\leq m\leq n\) are integers and \(\mu\) is a Borel measure on \(\mathbb{R}^n\) 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.
Comments
This is the main result from my PhD thesis and is the second paper I ever wrote (although it was the third published). It grew from the following result (which I started thinking about in the summer of 1993 whilst at a summer school in Finland and proved back in the UK in the autumn):
Suppose that \(\mu\) is a Borel measure on \(\mathbb{R}^2\) such that for \(\mu\)-almost every \(x\),Then \(\mu \) is \(1\)-rectifiable.
- The upper and lower \(1\)-densities of \(\mu\) at \(x\) are positive and finite.
- If \(\nu \) is a tangent measure of \(\mu \) at \(x\) then the support of \(\nu \) is connected.
In particular, if all the tangent measures have supports that are lines or half-lines, then the original measure is \(1\)-rectifiable.
If you have any queries about my work or would like a copy of this paper then please contact me at:
t.c.oneil@open.ac.uk