For any $x\in\mathbb{ R}^d$, $d\geq 2$, denote $\pi^x: \mathbb{ R}^d\backslash\{ x\}\rightarrow S^{ d-1}$ as the radial projection $$\pi^x(y)=\frac{ y-x}{ |y-x|}. $$ Given a Borel set $E\subset{ \Bbb R}^d$, $\dim_{ \mathcal{ H}} E\leq d-1$, in this paper we investigate for how many $x\in \mathbb{ R}^d$ the radial projection $\pi^x$ preserves the Hausdorff dimension of $E$, namely whether $\dim_{ \mathcal{ H}}\pi^x(E)=\dim_{ \mathcal{ H}} E$. We develop a general framework to link $\pi^x(E)$, $x\in F$ and $\pi^y(F)$, $y\in E$, for any Borel set $F\subset\mathbb{ R}^d$. In particular, whether $\dim_{ \mathcal{ H}}\pi^x(E)=\dim_{ \mathcal{ H}}E$ for some $x\in F$ can be reduced to whether $F$ is visible from some $y\in E$ (i.e. $\mathcal{ H}^{ d-1}(\pi^y(F))>0$). This allows us to apply Orponen's estimate on visibility to obtain $$\dim_{ \mathcal{ H}}\left\{ x\in\mathbb{ R}^d: \dim_{ \mathcal{ H}}\pi^x(E)<\dim_{ \mathcal{ H}}E\right\}\leq 2(d-1)-\dim_{ \mathcal{ H}}E,$$ for any Borel set $E\subset{ \Bbb R}^d$, $\dim_{ \mathcal{ H}} E\in(d-2, d-1]$. This improves the Peres-Schlag bound when $\dim_{ \mathcal{ H}} E\in(d-\frac{ 3}{ 2}, d-1]$, and it is optimal at the endpoint $\dim_{ \mathcal{ H}} E=d-1$.
Categories: math.CA math.CO math.MG