Projections of totally disconnected thin fractals with very thick shadows on ${\mathbb R}^d$

TL;DR

This paper investigates special fractals whose orthogonal projections match those of their convex hulls, providing criteria for self-affine sets and constructing fractals with projections as intervals within convex polytopes.

Contribution

It extends the Mastrand projection theorem to self-affine sets, establishing criteria for their projections and constructing fractals with prescribed projection properties within convex polytopes.

Findings

Fractals can have projections identical to their convex hulls.

Every convex polytope contains a totally disconnected fractal set with specific projection properties.

Constructed fractals project to intervals in every 1D subspace.

Abstract

We study an extreme scenario of the Mastrand projection theorem for which a fractal has the property that its orthogonal projection is the same as the orthogonal projection of its convex hull. We extend results in current literature and establish checkable criteria for self-affine sets to have such property. Using this, we show that every convex polytope on $\R^d$ contains a totally disconnected compact set, which is a union of self-affine sets, of dimension as close to 1 as possible, as well as a rectifiable 1-set, such that the fractal projects to an interval in every 1-dimensional subspace and its convex hull is the given polytope. Other convex sets and projections onto higher dimensional subspaces will also be discussed.