Metric projection
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. == Formal definition == 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.