# zbMATH — the first resource for mathematics

Selections of the metric projection operator and strict solarity of sets with continuous metric projection. (English. Russian original) Zbl 1426.41039
Sb. Math. 208, No. 7, 915-928 (2017); translation from Mat. Sb. 208, No. 7, 3-18 (2017).
For a normed linear space $$X$$ and a nonempty subset $$M$$, the metric projection $$P_M$$ is a set-valued mapping of $$X$$ to $$M$$ defined by $$P_Mx=\{y \in M\mid \inf _{z \in M}\|x-z\| =\|x-y\|\}$$ for $$x \in X$$. A selection of the metric projection is a single-valued mapping $$f : X \to M$$ satisfying $$f(x) \in P_Mx$$ for any $$x \in X$$. The set $$M$$ is called a sun if for any $$x \in X \setminus M$$ there exists $$y \in P_Mx$$ such that $$y \in P_M((1-\lambda)y+\lambda x)$$ for all $$\lambda\geq0$$, and $$M$$ is called an existence set if $$P_Mx \ne \varnothing$$ for any $$x \in X$$.
I. G. Tsar’kov [Math. Notes 47, No. 2, 218–227 (1990; Zbl 0703.46010); translation from Mat. Zametki 47, No. 2, 137–148 (1990)] proved that in a finite-dimensional Banach space an existence set with lower semicontinuous metric projection is a sun and has acyclic intersections with closed balls. In this paper, the author proves the following theorem: Let $$X$$ be a finite-dimensional Banach space with one of the following properties: (1) $$\dim X\leq 3$$; (2) $$X$$ is a (BM)-space introduced by A. L. Brown [Math. Ann. 279, No. 1–2, 87–101 (1987; Zbl 0607.41027)]; (3) $$X$$ is in the class (RBR) introduced by N. V. Nevesenko [Math. Notes 23, 308–312 (1978; Zbl 0408.41019)]. Then any closed set with lower semicontinuous metric projection in $$X$$ is a sun, admits a continuous selection of the metric projection, has contractible intersections with balls, and its (nonempty) intersection with any closed ball is a retract of this ball. The author also proves theorems on related properties.

##### MSC:
 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 54C65 Selections in general topology
Full Text:
##### References:
 [1] A. R. Alimov and I. G. Tsar’kov 2016 Connectedness and solarity in problems of best and near-best approximation Uspekhi Mat. Nauk71 1(427) 3-84 [2] English transl. in A. R. Alimov and I. G. Tsar’kov 2016 Russian Math. Surveys71 1 1-77 [3] Yu. V. Malykhin 2016 Relative widths of Sobolev classes in the uniform and integral metrics Function spaces, approximation theory and related areas in mathematical analysis Tr. Mat. Inst. Steklov. 293 MAIK, Moscow 217-223 [4] English transl. in Yu. V. Malykhin 2016 Proc. Steklov Inst. Math.293 209-215 [5] A. R. Alimov and I. G. Tsar’kov 2014 Connectedness and other geometric properties of suns and Chebyshev sets Fundam. Prikl. Mat.19 4 21-91 [6] English transl. in A. R. Alimov and I. G. Tsar’kov 2016 J. Math. Sci. (N. Y.)217 6 683-730 [7] L. P. Vlasov 1973 Approximative properties of sets in normed linear spaces Uspekhi Mat. Nauk28 6(174) 3-66 [8] English transl. in L. P. Vlasov 1973 Russian Math. Surveys28 6 1-66 [9] A. R. Alimov 2017 A monotone path connected set with metric projection which is radially lower semicontinuous is a strong sun Sibirsk. Mat. Zh.58 1 16-21 [10] A. R. Alimov 2012 Local solarity of suns in normed linear spaces Fundam. Prikl. Mat.17 7 3-14 [11] English transl. in A. R. Alimov 2014 J. Math. Sci. (N. Y.)197 4 447-454 [12] B. Brosowski and F. Deutsch 1974 Radial continuity of set-valued metric projections J. Approx. Theory11 3 236-253 [13] A. L. Brown, F. Deutsch, V. Indumathi and P. S. Kenderov 2002 Lower semicontinuity concepts, continuous selections, and set valued metric projections J. Approx. Theory115 1 120-143 [14] N. V. Nevesenko 1978 Strict sums and semicontinuity below metric projections in linear normed spaces Mat. Zametki23 4 563-572 [15] English transl. in N. V. Nevesenko 1978 Math. Notes23 4 308-312 [16] I. G. Tsar’kov 1990 Continuity of the metric projection, structural and approximate properties of sets Mat. Zametki47 2 137-148 [17] English transl. in I. G. Tsar’kov 1990 Math. Notes47 2 218-227 [18] F. Deutsch, W. Pollul and I. Singer 1973 On set-valued metric projections, Hahn-Banach extension maps, and spherical image maps Duke Math. J.40 2 355-370 [19] B. Brosowski and F. Deutsch 1972 Some new continuity concepts for metric projections Bull. Amer. Math. Soc.78 6 974-978 [20] D. Amir and F. Deutsch 1972 Suns, moons, and quasi-polyhedra J. Approx. Theory6 176-201 [21] B. Brosowski and F. Deutsch 1974 On some geometric properties of suns J. Approx. Theory10 3 245-267 [22] A. L. Brown 1987 Suns in normed linear spaces which are finite dimensional Math. Ann.279 1 87-101 [23] A. R. Alimov 2016 Mazur spaces and the 4.3-intersection property of $$(BM)$$-spaces Izv. Sarat. Univ. (N.S.) Ser. Mat. Mekh. Inform.16 2 133-137 [24] I. G. Tsar’kov 2016 Continuous $$\varepsilon$$-selection Mat. Sb.207 2 123-142 [25] English transl. in I. G. Tsar’kov 2016 Sb. Math.207 2 267-285 [26] A. R. Alimov 2006 Monotone path-connectedness of Chebyshev sets in the space $$C(Q)$$ Mat. Sb.197 9 3-18 [27] English transl. in A. R. Alimov 2006 Sb. Math.197 9 1259-1272 [28] Ş. Cobzaş 2013 Functional analysis in asymmetric normed spaces Front. Math. Birkhäuser/Springer Basel AG, Basel x+219 pp. [29] S. Park 1999 Ninety years of the Brouwer fixed point theorem Vietnam J. Math.27 3 182-222 [30] E. S. Polovinkin and M. V. Balashov 2004 Elements of convex and strongly convex analysis Fizmatlit, Moscow 416 pp. [31] A. R. Alimov 2014 Monotone path-connectedness and solarity of Menger-connected sets in Banach spaces Izv. Ross. Akad. Nauk Ser. Mat.78 4 3-18 [32] English transl. in A. R. Alimov 2014 Izv. Math.78 4 641-655 [33] U. H. Karimov and D. Repovš 2006 On the topological Helly theorem Topology Appl.153 10 1614-1621 [34] K. Borsuk 1967 Theory of retracts Monogr. Mat. 44 Państwowe Wydawnictwo Naukowe, Warsaw 251 pp. [35] K. Eda, U. H. Karimov and D. Repovš 2013 On 2-dimensional nonaspherical cell-like Peano continua: a simplified approach Mediterr. J. Math.10 1 519-528
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.