Product-space reformulation of convex feasibility
= Product-space reformulation of convex feasibility
Finding a point in $\bigcap_{j=1}^\ell C_j$ is equivalent to intersecting the product $C_1\times\cdots\times C_\ell$ with the diagonal subspace in $(\mathbb R^n)^\ell$. Projection onto the product is componentwise, while projection onto the diagonal replaces every component by their average.