Geodesic
Weakly monotone sets and continuous selection from a near-best approximation operator
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Harmonic analysis, approximation theory, and number theory, Tome 303 (2018), pp. 246-257

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A new notion of weak monotonicity of sets is introduced, and it is shown that an approximatively compact and weakly monotone connected (weakly Menger-connected) set in a Banach space admits a continuous additive (multiplicative) $\varepsilon $-selection for any $\varepsilon >0$. Then a notion of weak monotone connectedness (weak Menger connectedness) of sets with respect to a set of $d$-defining functionals is introduced. For such sets, continuous $(d^{-1},\varepsilon )$-selections are constructed on arbitrary compact sets. 
@article{TM_2018_303_a17,
     author = {I. G. Tsar'kov},
     title = {Weakly monotone sets and continuous selection from a near-best approximation operator},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {246--257},
     publisher = {mathdoc},
     volume = {303},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2018_303_a17/}
}
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I. G. Tsar'kov. Weakly monotone sets and continuous selection from a near-best approximation operator. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Harmonic analysis, approximation theory, and number theory, Tome 303 (2018), pp. 246-257. http://geodesic.mathdoc.fr/item/TM_2018_303_a17/