  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 $\dim_{\mathcal{H}} E=d-1$.
