A sharp exceptional set estimate for visibility

TL;DR

This paper establishes the optimal bound on the size of the exceptional set for visibility of Borel sets in Euclidean space, showing that sets with dimension greater than n-1 are visible from almost all points.

Contribution

It provides the sharp exceptional set estimate for visibility in Euclidean space, directly applying existing proof methods to achieve optimal bounds.

Findings

Exceptional set dimension bound is sharp for all n ≥ 2.

Sets with dimension > n-1 are visible from almost all points.

The proof method yields the optimal result without improving slicing theorems.

Abstract

A Borel set $B \subset \mathbb{R}^{n}$ is visible from $x \in \mathbb{R}^{n}$, if the radial projection of $B$ with base point $x$ has positive $\mathcal{H}^{n - 1}$ measure. I prove that if $\dim B > n - 1$, then $B$ is visible from every point $x \in \mathbb{R}^{n} \setminus E$, where $E$ is an exceptional set with dimension $\dim E \leq 2(n - 1) - \dim B$. This is the sharp bound for all $n \geq 2$. Many parts of the proof were already contained in a recent previous paper by P. Mattila and the author, where a weaker bound for $\dim E$ was derived as a corollary from a certain slicing theorem. Here, no improvement to the slicing result is obtained; in brief, the main observation of the present paper is that the proof method gives the optimal result, when applied directly to the visibility problem.