Approximative properties of sets and continuous selections

I. G. Tsar'kov

Sbornik. Mathematics, Tome 211 (2020) no. 8, pp. 1190-1211

Résumé

Sets admitting a continuous selection of the operators of best and near-best approximation are studied. Michael's classical continuous selection theorem is extended to the case of a lower semicontinuous metric projection in finite-dimensional spaces (with no a priori convexity conditions on its values). Sufficient conditions on the metric projection implying the solarity of the corresponding set are put forward in finite-dimensional polyhedral spaces. Available results for suns $V$ are employed to establish the existence of continuous selections of the relative (with respect to $V$) Chebyshev near-centre map and of the sets of relative (with respect to $V$) near-Chebyshev points in certain classical spaces. Bibliography: 30 titles.

Détail

Keywords: set-valued mapping, continuous selection, sun, monotone path-connected set, relative Chebyshev centre and point.

Bibliographie

[1] I. G. Tsar'kov, “Relations between certain classes of sets in Banach spaces”, Math. Notes, 40:2 (1986), 597–610 | DOI | MR | Zbl

[2] S. V. Konyagin, “O nepreryvnykh operatorakh obobschennogo ratsionalnogo priblizheniya”, Matem. zametki, 44:3 (1988), 404 | MR | Zbl

[3] I. G. Tsar'kov, “Properties of the sets that have a continuous selection from the operator $P^\delta$”, Math. Notes, 48:4 (1990), 1052–1058 | DOI | MR | Zbl

[4] I. G. Tsar'kov, “Properties of sets admitting stable $\varepsilon$-selections”, Math. Notes, 89:4 (2011), 572–576 | DOI | DOI | MR | Zbl

[5] K. S. Ryutin, “Uniform continuity of generalized rational approximations”, Math. Notes, 71:2 (2002), 236–244 | DOI | DOI | MR | Zbl

[6] E. D. Livshits, “On almost-best approximation by piecewise polynomial functions in the space $C[0,1]$”, Math. Notes, 78:4 (2005), 586–591 | DOI | DOI | MR | Zbl

[7] E. D. Livshits, “Stability of the operator of $\varepsilon$-projection to the set of splines in $C[0,1]$”, Izv. Math., 67:1 (2003), 91–119 | DOI | DOI | MR | Zbl

[8] E. D. Livshits, “Continuous selections of operators of almost best approximation by splines in the space $L_p[0,1]$. I”, Russ. J. Math. Phys., 12:2 (2005), 215–218 | MR | Zbl

[9] K. S. Ryutin, “Continuity of operators of generalized rational approximation in the space $L_1[0;1]$”, Math. Notes, 73:1 (2003), 142–147 | DOI | DOI | MR | Zbl

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

[11] 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

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

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

[14] 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

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

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

[17] I. G. Tsar'kov, “Continuous $\varepsilon$-selection and monotone path-connected sets”, Math. Notes, 101:6 (2017), 1040–1049 | DOI | DOI | MR | Zbl

[18] I. G. Tsar'kov, “Continuous selection for set-valued mappings”, Izv. Math., 81:3 (2017), 645–669 | DOI | DOI | MR | Zbl

[19] I. G. Tsar'kov, “Some applications of the geometric theory of approximations”, J. Math. Sci. (N.Y.), 245:1 (2020), 64–82 | DOI | MR

[20] I. G. Tsar'kov, “Continuous selections for metric projection operators and for their generalizations”, Izv. Math., 82:4 (2018), 837–859 | DOI | DOI | MR | Zbl

[21] I. G. Tsar'kov, “New criteria for the existence of a continuous $\varepsilon$-selection”, Math. Notes, 104:5 (2018), 727–734 | DOI | DOI | MR | Zbl

[22] I. G. Tsar'kov, “Continuous selections in asymmetric spaces”, Sb. Math., 209:4 (2018), 560–579 | DOI | DOI | MR | Zbl

[23] A. L. Brown, “Best $n$-dimensional approximation to sets of functions”, Proc. London Math. Soc. (3), 14:4 (1964), 577–594 | DOI | MR | Zbl

[24] E. Michael, “Continuous selections. I”, Ann. of Math. (2), 63:2 (1956), 361–382 | DOI | MR | Zbl

[25] 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

[26] L. Górniewicz, Topological fixed point theory of multivalued mappings, Topol. Fixed Point Theory Appl., 4, 2nd ed., Springer, Dordrecht, 2006, xiv+539 pp. | DOI | MR | Zbl

[27] K. Sakai, Geometric aspects of general topology, Springer Monogr. Math., Springer, Tokyo, 2013, xvi+521 pp. | DOI | MR | Zbl

[28] A. R. Alimov, “A monotone path connected Chebyshev set is a sun”, Math. Notes, 91:2 (2012), 290–292 | DOI | DOI | MR | Zbl

[29] A. R. Alimov, “Connectedness of suns in the space $c_0$”, Izv. Math., 69:4 (2005), 651–666 | DOI | DOI | MR | Zbl

[30] I. G. Tsarkov, “Ustoichivost otnositelnogo chebyshevskogo proektora v poliedralnykh prostranstvakh”, Tr. IMM UrO RAN, 24, no. 4, 2018, 235–245 | DOI | MR