Radial projections of rectifiable sets

TL;DR

The paper proves that for certain rectifiable sets in Euclidean space, if no m-plane contains almost all of the set, then a specific (m-1)-plane exists such that the radial projection from any outside point has positive measure.

Contribution

It establishes a new geometric property linking rectifiable sets and their radial projections relative to planes in Euclidean space.

Findings

Existence of a (m-1)-plane with positive measure radial projections

Connection between rectifiability and projection properties

Generalization of projection theorems for rectifiable sets

Abstract

We show that if no $m$-plane contains almost all of an $m$-rectifiable set $E \subset \R^{n}$, then there exists a single $(m - 1)$-plane $V$ such that the radial projection of $E$ has positive $m$-dimensional measure from every point outside $V$.