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renormings; non-reflexive Banach spaces; Chebyshev centers
Let $X$ be a non-reflexive real Banach space. Then for each norm $|\cdot|$ from a dense set of equivalent norms on $X$ (in the metric of uniform convergence on the unit ball of $X$), there exists a three-point set that has no Chebyshev center in $(X,|\cdot|)$. This result strengthens theorems by Davis and Johnson, van Dulst and Singer, and Konyagin.
[DJ] Davis W.J., Johnson W.B.: A renorming of non-reflexive Banach spaces. Israel J. Math. 14 (1973), 353-367. MR 0322481
[vDS] van Dulst D., Singer I.: On Kadec-Klee norms on Banach spaces. Studia Math. 54 (1976), 205-211. MR 0394132 | Zbl 0321.46012
[Ho] Holmes R.B.: A course in optimization and best approximation. Lecture Notes in Mathematics 257, Springer-Verlag, 1972. MR 0420367
[Ja] James R.C.: Reflexivity and the supremum of linear functionals. Ann. Math. 66 (1957), 159-169. MR 0090019 | Zbl 0079.12704
[Ko] Konyagin S.V.: A remark on renormings of nonreflexive spaces and the existence of a Chebyshev center. Moscow Univ. Math. Bull. 43 2 (1988), 55-56. MR 0938075
[Ve] Veselý L.: A geometric proof of a theorem about non-dual renormings. Proc. Amer. Math. Soc. 127 (1999), 2807-2809. MR 1670431
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