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 p_M, is the following set-valued function from X to M:

$p_M(x)=\arg \underset{y\in M}{\min }d(x,y)$

Equivalently:

<math>p_M(x) = \{y \in M : d(x,y) \leq d(x,y') \forall y'\in M \}

Chebyshev sets

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.

Continuity

Applications

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

External links

• Do projections onto convex sets always decrease distances?

References