Geodesic
On strong proximinality in normed linear spaces
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 70 (2016) no. 1.

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The paper deals with strong proximinality in normed linear spaces. It is proved that in  a compactly locally uniformly rotund Banach space, proximinality, strong proximinality, weak approximative compactness and  approximative compactness are all equivalent for closed convex sets. How strong proximinality can be transmitted to and from quotient spaces has also been discussed.
Keywords: Strongly proximinal set, approximatively compact set, strongly Chebyshev set, compactly locally uniformly rotund space 
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Gupta, Sahil; Narang, T. D. On strong proximinality in normed linear spaces. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 70 (2016) no. 1. http://geodesic.mathdoc.fr/item/AUM_2016_70_1_a5/

[1] Bandyopadhyay, Pradipta, Li, Yongjin, Lin, Bor-Luh, Narayana, Darapaneni, Proximinality in Banach spaces, J. Math. Anal. Appl. 341 (2008), 309-317.

[2] Cheney, E. W., Wulbert, D. E., The existence and uniqueness of best approximation, Math. Scand. 24 (1969), 113-140.

[3] Dutta, S., Shunmugraj, P., Strong proximinality of closed convex sets, J. Approx. Theory 163 (2011), 547-553.

[4] Effimov, N. V., Steckin, S. B., Approximative compactness and Chebyshev sets, Soviet Math. Dokl. 2 (1961), 1226-1228.

[5] Finet, C., Quarta, L., Some remarks on M-ideals and strong proximinality, Bull. Korean Math. Soc. 40 (2003), 503-508.

[6] Godefroy, G., Indumathi, V., Strong proximinality and polyhedral spaces, Rev. Mat. Complut. 14 (2001), 105-125.

[7] Jayanarayanan, C. R., Paul, T., Strong proximinality and intersection properties of balls in Banach spaces, J. Math. Anal. Appl. 426 (2015), 1217-1231.

[8] Narayana, D., Strong proximinality and renorming, Proc. Amer. Math. Soc. 134 (2005), 1167-1172.

[9] Panda, B. B., Kapoor, O. P., A generalization of the local uniform rotundity of the norm, J. Math. Anal. Appl. 52 (1975), 300-308.

[10] Singer, I., Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Springer-Verlag, New York, 1967.

[11] Vlasov, L. P., The concept of approximative compactness and its variants, Mat. Zametki 16 (1974), 337-348 (Russian), English transl. in Math. Notes 16, No. 2 (1974), 786-792.

[12] Zhang, Z. H., Shi, Z. R., Convexities and approximative compactness and continuity of metric projection in Banach spaces, J. Approx. Theory 161 (2009), 802-812.