If 𝒮\mathcal{S} is convex, then for any 𝐲∈𝒮\mathbf{y} \in \mathcal{S},

||𝐲−𝖯𝒮(𝐱)||2≤||𝐲−𝐱||2||\mathbf{y}-\mathsf{P}_\mathcal{S}(\mathbf{x})||_2 \leq ||\mathbf{y}-\mathbf{x}||_2

See: projection oracle