Geodesic
Selections of the best and near-best approximation operators and solarity
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Harmonic analysis, approximation theory, and number theory, Tome 303 (2018), pp. 17-25

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In a finite-dimensional Banach space, a closed set with lower semicontinuous metric projection is shown to have a continuous selection of the near-best approximation operator. Such a set is known to be a sun. In the converse question of the stability of best approximation by suns, it is proved that a strict sun in a finite-dimensional Banach space of dimension at most $3$ is a $P$-sun, has a contractible set of nearest points, and admits a continuous $\varepsilon $-selection from the operator of near-best approximation for any $\varepsilon >0$. A number of approximative and geometric properties of sets with lower semicontinuous metric projection are obtained.
Keywords: lower semicontinuity of the metric projection, selection of the metric projection, sun, strict sun, near-best approximation. 
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     author = {A. R. Alimov},
     title = {Selections of the best and near-best approximation operators and solarity},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {17--25},
     publisher = {mathdoc},
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     url = {http://geodesic.mathdoc.fr/item/TM_2018_303_a1/}
}
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A. R. Alimov. Selections of the best and near-best approximation operators and solarity. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Harmonic analysis, approximation theory, and number theory, Tome 303 (2018), pp. 17-25. http://geodesic.mathdoc.fr/item/TM_2018_303_a1/