  Given a set in the plane, the average length of its projections over all
directions is called Favard length. This quantity measures the size of a set,
and is closely related to metric and geometric properties of the set such as
rectifiability, Hausdorff dimension, and analytic capacity. In this paper, we
develop new geometric techniques for estimating Favard length. We will give a
short geometrically motivated proof relating Hausdorff dimension to the decay
rate of the Favard length of neighborhoods of a set. We will also show that the
sequence of Favard lengths of the generations of a self-similar set is convex;
this has direct applications to giving lower bounds on Favard length for
various fractal sets.
