The proximum is a term used mainly in numerical mathematics from the theory of metric spaces . The Proximum to a point within a not containing amount is from that point , to be the minimum distance from.
${\ displaystyle x}$${\ displaystyle x}$${\ displaystyle Y}$${\ displaystyle Y}$${\ displaystyle x}$

Be a metric space , a subset, and arbitrary. The distance between the element and the subset is defined by
means of the distance function${\ displaystyle (X, d)}$${\ displaystyle Y \ subset X}$${\ displaystyle x \ in X}$${\ displaystyle x}$${\ displaystyle Y}$${\ displaystyle \ operatorname {dist}}$

${\ displaystyle \ operatorname {dist} (x, Y): = \ inf _ {y \ in Y} d (x, y) \ ,.}$

this is called the proximum or best approximation to in .
${\ displaystyle p}$${\ displaystyle x}$${\ displaystyle Y}$

If there is a proximum, it need not be unique.

Usually one has to do with a normalized space in approximation theory . A proximum to in is then - if it exists - characterized by the equation
${\ displaystyle (X, \ lVert \ cdot \ rVert)}$${\ displaystyle p}$${\ displaystyle x \ in X}$${\ displaystyle Y \ subset X}$

${\ displaystyle \ lVert xp \ rVert = \ inf _ {y \ in Y} \ lVert xy \ rVert}$

To the existence of a proximum

Be a metric space . be a compact subset. Then each has a proximum in .${\ displaystyle (X, d) \,}$${\ displaystyle A \ subset X}$${\ displaystyle x \ in X}$${\ displaystyle A}$

Be a normalized space. be a finite-dimensional subspace and a closed subset. Then each has a proximum in .${\ displaystyle (X, \ lVert \ cdot \ rVert)}$${\ displaystyle V \ subset X}$${\ displaystyle Y \ subset V}$${\ displaystyle x \ in X}$${\ displaystyle Y}$

Uniqueness of the proximal in Chebyshev systems

Be a Chebyshev system . Then the proximum for off is uniquely determined.
${\ displaystyle f \ in C [a, b], U \ subset C [a, b]}$${\ displaystyle f}$${\ displaystyle U}$

Let be a finite-dimensional subspace of . If the proximum of each is clearly determined, then there is a Chebyshev system .
${\ displaystyle U}$${\ displaystyle C [a, b]}$${\ displaystyle f \ in C [a, b]}$${\ displaystyle U}$${\ displaystyle U}$

Alternant criterion in Chebyshev systems

Be a -dimensional Chebyshev system . is a proximum for out if and only if there are places with such that
${\ displaystyle f \ in C [a, b], U \ subset C [a, b]}$${\ displaystyle n}$${\ displaystyle u_ {0} \ in U}$${\ displaystyle f}$${\ displaystyle U}$${\ displaystyle n + 1}$${\ displaystyle x_ {i}}$${\ displaystyle a \ leq x_ {0} <x_ {1} <\ cdots <x_ {n} \ leq b}$

${\ displaystyle | f (x_ {i}) - u_ {0} (x_ {i}) | = \ max _ {x \ in [a, \, b]} | f (x) -u_ {0} ( x) |}$, (Extreme point)${\ displaystyle i = 0, \ ldots, n}$

This follows from the Kolmogorow criterion from approximation theory . The Remez algorithm for the numerical determination of the proximum in Chebyshev systems is based on this criterion .

Proximum in the Hilbert space

If a Hilbert space and a closed convex non-empty subset, then the proximum is unique, that is, there is exactly one with
for each${\ displaystyle X}$${\ displaystyle Y \ subset X}$${\ displaystyle x \ in X}$${\ displaystyle p \ in Y}$

${\ displaystyle \ lVert xp \ rVert \ leq \ lVert xy \ rVert \, \, \ forall \, y \ in Y}$.

If a closed sub-vector space , the proximum is obtained as an orthogonal projection of on .
${\ displaystyle Y}$${\ displaystyle p}$${\ displaystyle x}$${\ displaystyle Y}$