In mathematics, a metric projection is a function that maps each element of a metric space to the set of points nearest to that element in some fixed sub-space.^[1][2]

Formally, let X be a metric space with distance metric d, and let M be a fixed subset of X. Then the metric projection associated with M, denoted p_M, is the following set-valued function from X to M:

${\displaystyle p_{M}(x)=\arg \min _{y\in M}d(x,y)}$

Equivalently:

${\displaystyle p_{M}(x)=\{y\in M:d(x,y)\leq d(x,y')\forall y'\in M\}=\{y\in M:d(x,y)=d(x,M)\}}$

The elements in the set ${\displaystyle \arg \min _{y\in M}d(x,y)}$ are also called elements of best approximation. This term comes from constrained optimization: we want to find an element nearer to x, under the constraint that the solution must be a subset of M. The function p_M is also called an operator of best approximation.^{[citation needed]}

In general, p_M is set-valued, as for every x, there may be many elements in M that have the same nearest distance to x. In the special case in which p_M is single-valued, the set M is called a Chebyshev set. As an example, if (X,d) is a Euclidean space (Rⁿ with the Euclidean distance), then a set M is a Chebyshev set if and only if it is closed and convex.^[3]

If M is non-empty compact set, then the metric projection p_M is upper semi-continuous, but might not be lower semi-continuous. But if X is a normed space and M is a finite-dimensional Chebyshev set, then p_M is continuous.^{[citation needed]}

Moreover, if X is a Hilbert space and M is closed and convex, then p_M is Lipschitz continuous with Lipschitz constant 1.^{[citation needed]}

Metric projections are used both to investigate theoretical questions in functional analysis and for practical approximation methods.^[4] They are also used in constrained optimization.^[5]

• Do projections onto convex sets always decrease distances?