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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.


Be a metric space , a subset, and arbitrary. The distance between the element and the subset is defined by means of the distance function

Now exists with:

this is called the proximum or best approximation to in .

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

To the existence of a proximum

  • Be a metric space . be a compact subset. Then each has a proximum in .
  • Be a normalized space. be a finite-dimensional subspace and a closed subset. Then each has a proximum in .

Uniqueness of the proximal in Chebyshev systems

Be a Chebyshev system . Then the proximum for off is uniquely determined.

Let be a finite-dimensional subspace of . If the proximum of each is clearly determined, then there is a Chebyshev system .

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

  • , (Extreme point)
  • , (alternating)

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


If a closed sub-vector space , the proximum is obtained as an orthogonal projection of on .

See also