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 pM, is the following set-valued function from X to M:
p M ( x ) = arg min y ∈ M d ( x , y ) {\displaystyle p_{M}(x)=\arg \min _{y\in M}d(x,y)}
Equivalently:
p M ( x ) = { y ∈ M : d ( x , y ) ≤ d ( x , y ′ ) ∀ y ′ ∈ M } = { y ∈ M : d ( x , y ) = d ( x , M ) } {\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 arg min y ∈ M d ( x , y ) {\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 pM is also called an operator of best approximation.[citation needed]
In general, pM 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 pM 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 pM 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 pM is continuous.[citation needed]
Moreover, if X is a Hilbert space and M is closed and convex, then pM 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]