Blaschke selection theorem

The sampling rate of Blaschke (English Blaschke Selection Theorem ) is a mathematical theorem, which deals with a convergence problem of convex geometry. The phrase is attributed to the transition field between convex geometry and topology. He was introduced by the surveyor Wilhelm Blaschke in his writing circle and sphere in 1916.

Formulation of the selection set

The sampling rate of Blaschke can be in modern version as formulate follows:

Other formulation of the selection set

If we denote by the class of sets of non-empty compact convex subsets of normed vector space with the Hausdorff metric, so says the selection set:

Applications

The sampling rate is often used where existence proofs are to lead to the extremal convex geometry. As Wilhelm Blaschke already shows in the circle and sphere, can be derived using the selection set, for example, the isoperimetric inequality.

Related results

The sampling rate of Blaschke leads to the conclusion from the set of Arzelà - Ascoli and proves in a generalized version to that ( in the classical form ) even as equivalent.