Continuous selections for metric projection operators and for their generalizations. (English. Russian original) Zbl 1410.46007
Izv. Math. 82, No. 4, 837-859 (2018); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 82, No. 4, 199-224 (2018).
For Hausdorff-continuous metric projections onto an existence set \(M\) in a Banach space with values fulfilling special conditions concerning infinite connectivity it is proved that the values of such projections must all be contractible, and that \(M\) admits a continuous selection for the metric projection. Conditions under which \(M\) has a continuous \(\varepsilon\)-selection (with respect to an asymmetric seminorm) are then formulated.
For finite-dimensional asymmetric Banach spaces in which the unit ball is a convex polyhedron and for existence sets \(M\) admitting a lower semicontinuous metric projection, there exists a continuous selection for the metric projection. This affirmatively answers a question posed by {A. L. Brown}. Applications in approximation theory and the calculus of variations are mentioned.
For finite-dimensional asymmetric Banach spaces in which the unit ball is a convex polyhedron and for existence sets \(M\) admitting a lower semicontinuous metric projection, there exists a continuous selection for the metric projection. This affirmatively answers a question posed by {A. L. Brown}. Applications in approximation theory and the calculus of variations are mentioned.
Reviewer: Włodzimierz Ślęzak (Bydgoszcz)
MSC:
| 46B20 | Geometry and structure of normed linear spaces |
| 41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |
| 54C65 | Selections in general topology |
| 49J53 | Set-valued and variational analysis |
Keywords:
asymmetric spaces; semimetric space; infinitely connected set; existence set; \(\varepsilon\)-selection; polyhedral space; strict solarity point; metric projectionReferences:
| [1] | Alimov, A. R.; Tsar’kov, I. G., Connectedness and other geometric properties of suns and Chebyshev sets, Fundam. i Prikl. Mat., 19, 4, 21-91, (2014) · Zbl 1361.46012 · doi:10.1007/s10958-016-3000-1 |
| [2] | Alimov, A. R.; Tsar’kov, I. G., Connectedness and solarity in problems of best and near-best approximation, Uspekhi Mat. Nauk, 71, 1-427, 3-84, (2016) · Zbl 1350.41031 · doi:10.4213/rm9698 |
| [3] | Konyagin, S. V., On continuous operators of generalized rational approximation, Mat. Zametki, 44, 3, (1988) · Zbl 0694.41022 |
| [4] | Livshits, E. D., Stability of the operator of, Izv. Ross. Akad. Nauk Ser. Mat., 67, 1, 99-130, (2003) · Zbl 1079.41007 · doi:10.4213/im420 |
| [5] | Ryutin, K. S., Continuity of operators of generalized rational approximation in the space, Mat. Zametki, 73, 1, 148-153, (2003) · Zbl 1024.41012 · doi:10.4213/mzm609 |
| [6] | Ryutin, K. S., Uniform continuity of generalized rational approximations, Mat. Zametki, 71, 2, 261-270, (2002) · Zbl 1023.41011 · doi:10.4213/mzm345 |
| [7] | Tsar’kov, I. G., Local and global continuous, Izv. Ross. Akad. Nauk Ser. Mat., 80, 2, 165-184, (2016) · Zbl 1356.46013 · doi:10.4213/im8348 |
| [8] | Tsar’kov, I. G., Continuous, Mat. Sb., 207, 2, 123-142, (2016) · Zbl 1347.41047 · doi:10.4213/sm8481 |
| [9] | Tsar’kov, I. G., Properties of the sets that have a continuous selection from the operator, Mat. Zametki, 48, 4, 122-131, (1990) · Zbl 0729.46011 · doi:10.1007/BF01139608 |
| [10] | Tsar’kov, I. G., Properties of sets admitting stable, Mat. Zametki, 89, 4, 608-613, (2011) · Zbl 1245.49024 · doi:10.4213/mzm9101 |
| [11] | Michael, E., Continuous selections. I, Ann. of Math. (2), 63, 2, 361-382, (1956) · Zbl 0071.15902 · doi:10.2307/1969615 |
| [12] | Tsar’kov, I. G., Relations between certain classes of sets in Banach spaces, Mat. Zametki, 40, 2, 174-196, (1986) · Zbl 0625.46017 · doi:10.1007/BF01159114 |
| [13] | Tsar’kov, I. G., Continuous selection for set-valued mappings, Izv. Ross. Akad. Nauk Ser. Mat., 81, 3, 189-216, (2017) · Zbl 1376.54021 · doi:10.4213/im8450 |
| [14] | Alimov, A. R., Monotone path-connectedness of Chebyshev sets in the space, Mat. Sb., 197, 9, 3-18, (2006) · Zbl 1147.41011 · doi:10.4213/sm1129 |
| [15] | Tsar’kov, I. G., Continuous, Mat. Zametki, 101, 6, 919-931, (2017) · Zbl 1376.41028 · doi:10.4213/mzm11342 |
| [16] | Górniewicz, L., Topol. Fixed Point Theory Appl., 4, (2006), Springer: Springer, Dordrecht · Zbl 1107.55001 · doi:10.1007/1-4020-4666-9 |
| [17] | Tsar’kov, I. G., Some applications of geometric approximation theory, Diff. uravneniya. Mat. analiz, 143, 63-80, (2017) |
| [18] | Tsar’kov, I. G., Continuity of the metric projection, structural and approximate properties of sets, Mat. Zametki, 47, 2, 137-148, (1990) · Zbl 0695.46003 · doi:10.1007/BF01156834 |
| [19] | Sakai, K., Springer Monogr. Math., (2013), Springer: Springer, Tokyo · Zbl 1280.54001 · doi:10.1007/978-4-431-54397-8 |
| [20] | Balaganskii, V. S.; Vlasov, L. P., The problem of convexity of Chebyshev sets, Uspekhi Mat. Nauk, 51, 6-312, 125-188, (1996) · Zbl 0931.41017 · doi:10.4213/rm1020 |
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