Geometric Bounds for Favard Length

TL;DR

This paper introduces new geometric methods to estimate Favard length, linking it to fractal properties and providing bounds for self-similar sets, enhancing understanding of geometric measure theory.

Contribution

It develops novel geometric techniques to estimate Favard length and establishes convexity of Favard length sequences for self-similar sets, with applications to fractal geometry.

Findings

A geometric proof relating Hausdorff dimension to Favard length decay.

Favard length sequences of self-similar sets are convex.

Lower bounds on Favard length for fractal sets are derived.

Abstract

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.