Geodesic
New Criteria for the Existence of a Continuous $\varepsilon$-Selection
Matematičeskie zametki, Tome 104 (2018) no. 5, pp. 745-754

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We study sets admitting a continuous selection of near-best approximations and characterize those sets in Banach spaces for which there exists a continuous $\varepsilon$-selection for each $\varepsilon>0$. The characterization is given in terms of $P$-cell-likeness and similar properties. In particular, we show that a closed uniqueness set in a uniformly convex space admits a continuous $\varepsilon$-selection for each $\varepsilon>0$ if and only if it is $\mathring{B}$-approximately trivial. We also obtain a fixed point theorem.
Keywords: continuous $\varepsilon$-selection, fixed point. 
@article{MZM_2018_104_5_a11,
     author = {I. G. Tsar'kov},
     title = {New {Criteria} for the {Existence} of a {Continuous} $\varepsilon${-Selection}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {745--754},
     publisher = {mathdoc},
     volume = {104},
     number = {5},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2018_104_5_a11/}
}
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I. G. Tsar'kov. New Criteria for the Existence of a Continuous $\varepsilon$-Selection. Matematičeskie zametki, Tome 104 (2018) no. 5, pp. 745-754. http://geodesic.mathdoc.fr/item/MZM_2018_104_5_a11/