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For a nonempty closed convex subset $C$ of a finite-dimensional inner-product space, ${\Pi }_C(x)$ is the unique point of $C$ minimizing $\Vert x-z\Vert$.

This closest-point map is the metric projection onto a closed convex set. Repeated projections onto several sets form the separate projections onto convex sets method.

Projections onto convex sets is an iterative feasibility method that successively applies a Euclidean projection onto a convex set to each closed convex constraint set. With two sets it uses $x_{n+1}=P_D(P_C(x_n))$. It seeks a point in the intersection; a single projection onto one set is a different operation, and the method does not generally compute the closest intersection point to the initial iterate.