On the dimension of visible parts

TL;DR

This paper proves that for most directions, the visible parts of a compact set in higher-dimensional space have Hausdorff dimension slightly less than the ambient space dimension.

Contribution

It establishes a new upper bound on the Hausdorff dimension of visible parts of compact sets in Euclidean spaces for almost every viewing direction.

Findings

Visible parts have Hausdorff dimension at most n - 1/(50n) in almost every direction.

The result improves understanding of the geometric complexity of visible parts.

The proof applies to all compact sets in Euclidean space with dimension n ≥ 2.

Abstract

I prove that the visible parts of a compact set in $\mathbb{R}^{n}$, $n \geq 2$, have Hausdorff dimension at most $n - \tfrac{1}{50n}$ from almost every direction.