Continuous selection from the sets of best and near-best approximation

Selections of the Best and Near-Best Approximation Operators and Solarity

“…The basic definitions will be recalled below. In the present paper, we continue the investigation started in [4][5][6][7][8]22].…”

“…In view of Tsar'kov's recent result on local selections (see [20;22,Theorem 4]), this implies that the set M admits a continuous additive (multiplicative) ε-selection for any ε > 0. Now by Tsar'kov's…”

“…It is well known [16] that a planar compact set is acyclic if and only if it is cell-like. Correspondingly, a P -acyclic set M in a space of dimension n ≤ 3 is P -cell-like, and hence, by the above Shchepin-Brodskii theorem on approximation of a mapping with cell-like values by a continuous single-valued mapping, M admits a continuous ε-selection for any ε > 0 (here one can also invoke [22,Theorem 4]). This, in view of a recent result by Tsar'kov [22], shows that the set M is B-cell-like.…”

“…Let M be a subset of a finite-dimensional Banach space. According to a result by Tsar'kov [22,Theorem 5], if a set M in a finite-dimensional space has lower semicontinuous metric projection, then M is B-cell-like. In view of another result by Tsar'kov [22,Theorem 2], this implies that M admits a continuous ε-selection for all ε > 0.…”

Weakly Monotone Sets and Continuous Selection from a Near-Best Approximation Operator

“…Various results on ε-selections for specific function sets can be found in [7-10, 12, 13, 15-17]. The general properties of such sets were studied and considered in [1,[3][4][5][19][20][21][22].…”