Non-existence of multi-line Besicovitch sets

TL;DR

This paper proves that a compact set in the plane containing many line segments in various directions must have positive measure, highlighting a fundamental geometric property of such sets.

Contribution

It establishes the non-existence of multi-line Besicovitch sets with zero measure, advancing understanding of geometric measure theory.

Findings

Sets with many line segments in different directions have positive measure

No compact zero-measure set can contain a positive-dimensional family of line segments in many directions

Supports the conjecture that certain geometric configurations imply positive measure

Abstract

If a compact set K \subset R^2 contains a positive-dimensional family of line-segments in positively many directions, then K has positive measure.