  We study an extension of the Falconer distance problem in the multiparameter
setting. Given $\ell\geq 1$ and $\mathbb{R}^{d}=\mathbb{R}^{d_1}\times\cdots
\times\mathbb{R}^{d_\ell}$, $d_i\geq 2$. For any compact set $E\subset
\mathbb{R}^{d}$ with Hausdorff dimension larger than
$d-\frac{\min(d_i)}{2}+\frac{1}{4}$ if $\min(d_i) $ is even,
$d-\frac{\min(d_i)}{2}+\frac{1}{4}+\frac{1}{4\min(d_i)}$ if $\min(d_i) $ is
odd, we prove that the multiparameter distance set of $E$ has positive
$\ell$-dimensional Lebesgue measure. A key ingredient in the proof is a new
multiparameter radial projection theorem for fractal measures.
