# A local version of the Projection Theorem

## 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.

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 $$x$$,
• 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.
Then $$\mu$$ is $$1$$-rectifiable.

In particular, if all the tangent measures have supports that are lines or half-lines, then the original measure is $$1$$-rectifiable.