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. 
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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/

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