  We provide several new answers on the question: how do radial projections
distort the dimension of planar sets? Let $X,Y \subset \mathbb{R}^{2}$ be
non-empty Borel sets. If $X$ is not contained on any line, we prove that \[
\sup_{x \in X} \dim_{\mathrm{H}} \pi_{x}(Y) \geq \min\{\dim_{\mathrm{H}}
X,\dim_{\mathrm{H}} Y,1\}. \] If $\dim_{\mathrm{H}} Y > 1$, we have the
following improved lower bound: \[ \sup_{x \in X} \dim_{\mathrm{H}} \pi_{x}(Y
\, \setminus \, \{x\}) \geq \min\{\dim_{\mathrm{H}} X + \dim_{\mathrm{H}} Y -
1,1\}. \] Our results solve conjectures of Lund-Thang-Huong, Liu, and the first
author. Another corollary is the following continuum version of Beck's theorem
in combinatorial geometry: if $X \subset \mathbb{R}^{2}$ is a Borel set with
the property that $\dim_{\mathrm{H}} (X \, \setminus \, \ell) =
\dim_{\mathrm{H}} X$ for all lines $\ell \subset \mathbb{R}^{2}$, then the line
set spanned by $X$ has Hausdorff dimension at least $\min\{2\dim_{\mathrm{H}}
X,2\}$.
  While the results above concern $\mathbb{R}^{2}$, we also derive some
counterparts in $\mathbb{R}^{d}$ by means of integralgeometric considerations.
The proofs are based on an $\epsilon$-improvement in the Furstenberg set
problem, due to the two first authors, a bootstrapping scheme introduced by the
second and third author, and a new planar incidence estimate due to Fu and Ren.
