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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.

For a nonempty closed convex set $C$ in a Hilbert space, its metric projection onto a closed convex set obeysThis stronger inequality implies nonexpansiveness of metric projection, $\Vert {\Pi }_C(x)-{\Pi }_C(y)\Vert \le \Vert x-y\Vert$, by the Cauchy-Schwarz inequality.

Put $p={\Pi }_C(x)$ and $q={\Pi }_C(y)$. The variational characterization of convex projection gives $⟨x-p,q-p⟩\le 0$ and $⟨y-q,p-q⟩\le 0$. Adding the corresponding inequalities in the common direction $p-q$ yields $⟨x-y-(p-q),p-q⟩\ge 0$, which is the displayed estimate.

Thus projection cannot magnify errors in the input, a useful stability property for projected gradient descent and projections onto convex sets. It need not be a strict contraction: projecting two points already in $C$ leaves both unchanged. The closedness, convexity and inner-product geometry are essential to this nearest-point theorem.

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.