= Solution
Work in the product space $(\mathbb R^n)^\ell$ and define
$$
\mathcal C=C_1\times\cdots\times C_\ell,
\qquad
\mathcal D=\{(x,\ldots,x):x\in\mathbb R^n\}.
$$
Then $\mathcal C\cap\mathcal D$ is nonempty exactly when $\bigcap_jC_j$ is nonempty. For $X=(x_1,\ldots,x_\ell)$,
$$
\boxed{P_{\mathcal C}X=(P_{C_1}x_1,\ldots,P_{C_\ell}x_\ell),}
$$
while, with $\bar x=\ell^{-1}\sum_jx_j$,
$$
\boxed{P_{\mathcal D}X=(\bar x,\ldots,\bar x).}
$$
This is the <product-space reformulation of convex feasibility>.
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