Geodesic
Selections of the best and near-best approximation operators and solarity
Informatics and Automation, 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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A. R. Alimov. Selections of the best and near-best approximation operators and solarity. Informatics and Automation, Harmonic analysis, approximation theory, and number theory, Tome 303 (2018), pp. 17-25. http://geodesic.mathdoc.fr/item/TRSPY_2018_303_a1/

[1] A. R. Alimov, “Monotone path-connectedness of Chebyshev sets in the space $C(Q)$”, Sb. Math., 197:9 (2006), 1259–1272 | DOI | DOI | MR | Zbl

[2] A. R. Alimov, “Preservation of approximative properties of Chebyshev sets and suns in a plane”, Moscow Univ. Math. Bull., 63:5 (2008), 198–201 | DOI | MR | Zbl

[3] A. R. Alimov, “Local solarity of suns in normed linear spaces”, J. Math. Sci., 197:4 (2014), 447–454 | DOI | MR | Zbl

[4] A. R. Alimov, “Monotone path-connectedness and solarity of Menger-connected sets in Banach spaces”, Izv. Math., 78:4 (2014), 641–655 | DOI | DOI | MR | Zbl

[5] A. R. Alimov, “Selections of the metric projection operator and strict solarity of sets with continuous metric projection”, Sb. Math., 208:7 (2017), 915–928 | DOI | DOI | MR | Zbl

[6] Alimov A. R., “Continuity of the metric projection and local solar properties of sets”, Set-Valued Var. Anal., 2017 | DOI

[7] A. R. Alimov, “A monotone path-connected set with outer radially lower continuous metric projection is a strict sun”, Sib. Math. J., 58:1 (2017), 11–15 | DOI | MR | MR | Zbl

[8] A. R. Alimov, E. V. Shchepin, “Convexity of Chebyshev sets with respect to tangent directions”, Russ. Math. Surv., 73:2 (2018), 366–368 | DOI | DOI | MR | Zbl

[9] A. R. Alimov, I. G. Tsar'kov, “Connectedness and other geometric properties of suns and Chebyshev sets”, J. Math. Sci., 217:6 (2016), 683–730 | DOI | MR | Zbl

[10] A. R. Alimov, I. G. Tsar'kov, “Connectedness and solarity in problems of best and near-best approximation”, Russ. Math. Surv., 71:1 (2016), 1–77 | DOI | DOI | MR | Zbl

[11] Blatter J., Morris P. D., Wulbert D. E., “Continuity of the set-valued metric projection”, Math. Ann., 178:1 (1968), 12–24 | DOI | MR | Zbl

[12] S. A. Bogatyi, “Topological Helly theorem”, Fundam. Prikl. Mat., 8:2 (2002), 365–405 | MR | Zbl

[13] Cobzaş Ş., Functional analysis in asymmetric normed spaces, Front. Math., Birkhäuser, Basel, 2013 | DOI | MR | Zbl

[14] Górniewicz L., Topological fixed point theory of multivalued mappings, Springer, Dordrecht, 2006 | MR | Zbl

[15] Kamal A., “On proximinality and sets of operators. I: Best approximation by finite rank operators”, J. Approx. Theory, 47 (1986), 132–145 | DOI | MR | Zbl

[16] Karimov U. H., Repovš D., “On the topological Helly theorem”, Topology Appl., 153:10 (2006), 1614–1621 | DOI | MR | Zbl

[17] S. V. Konyagin, “On continuous operators of generalized rational approximation”, Mat. Zametki, 44:3 (1988), 404 | MR | Zbl

[18] E. V. Shchepin, N. B. Brodskii, “Selections of filtered multivalued mappings”, Proc. Steklov Inst. Math., 212 (1996), 209–229 | MR | Zbl

[19] I. G. Tsar'kov, “Continuous $\varepsilon $-selection”, Sb. Math., 207:2 (2016), 267–285 | DOI | DOI | MR | Zbl

[20] I. G. Tsar'kov, “Local and global continuous $\varepsilon $-selection”, Izv. Math., 80:2 (2016), 442–461 | DOI | DOI | MR | Zbl

[21] I. G. Tsar'kov, “Some applications of geometric theory of approximations”, Differential Equations. Mathematical Analysis, Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz., 143, VINITI, M., 2017, 63–80

[22] I. G. Tsar'kov, “Continuous selection from the sets of best and near-best approximation”, Dokl. Math., 96:1 (2017), 362–364 | DOI | MR | Zbl

[23] L. P. Vlasov, “Approximative properties of sets in normed linear spaces”, Russ. Math. Surv., 28:6 (1973), 1–66 | DOI | MR | Zbl | Zbl

[24] Wegmann R., “Some properties of the peak-set-mapping”, J. Approx. Theory, 8:3 (1973), 262–284 | DOI | MR | Zbl