Конструктивное Доказательство Теоремы Киршбрауна

A constructive version of the extremum value theorem for spaces of vector-valued functions

“…This theorem was first addressed by Kirszbraun [7] and Valentine [12], and then revisited by Brehm [4]. A similar proof can be found in Akopyan and Tarasov [1] and Petrunin and Yashinski [8, page 21]. In our restricted case, we consider only points with rational coordinates.…”

“…At this step, we may take a closed ballB( p j 2 , r 2 ) that containsB( p j 2 , r 1 ), and in turn f (x 2 ), with a radius r 2 that is larger than r 1 by a value that depends only on the choice of the initial mesh step δ . Since the total number of steps is fixed by N , setting δ sufficiently small ensures that all ψ(X 0 ) are within 1 lN from the respective f (X 0 ) while preserving the Lipschitz constant. Now, we need to extend ψ to the whole X .…”

“…In this work, we address this problem and prove the extremum value theorem in an approximate format constructively. 1 As the underlying metric space, we take spaces of functions from an arbitrarily dimensioned Euclidean space to the plane. The theorem is stated as follows: Theorem 1.2 Let F be the space of all uniformly Lipschitz and uniformly bounded functions from a closed hypercubeH(b, K) ⊂ R n to the plane R 2 , and let J be a uniformly continuous functional from F to R. Then, for any k ∈ N, there exists an…”