| # Euclidean projection on a set An Euclidean projection of a point on a set is a point that achieves the smallest Euclidean distance from to the set. That is, it is any solution to the optimization problem When the set is convex, there is a unique solution to the above problem. In particular, the projection on an affine subspace is unique. Example: assume that is the hyperplane The projection problem reads as a linearly constrained least-squares problem, of particularly simple form: The projection of on turns out to be aligned with the coefficient vector . Indeed, components of orthogonal to don't appear in the constraint, and only increase the objective value. Setting in the equation defining the hyperplane and solving for the scalar we obtain , so that the projection is .