On the size of the set of points where the metric projection exists. (English) Zbl 1067.46015

A closed convex set \(C\) in a reflexive, locally uniformly convex Banach space \(X\) admits best approximations to a dense \(G_{\delta}\) subset of \(X\) [E. Asplund, Isr. J. Math. 4, 213–216 (1966; Zbl 0143.34904)]. This work answers negatively a conjecture that if the hypothesis of the local uniform convexity is reduced to strict convexity then almost every \(x \in X\) has a best approximation in \(C\). A similar negative result is presented for farthest points and a conjecture that a property of differentials in reflexive spaces remains true for a non-reflexive space is shown to be false.


46B20 Geometry and structure of normed linear spaces
41A50 Best approximation, Chebyshev systems
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
Full Text: DOI


[1] Asplund, E., Farthest points in reflexive locally uniformly rotund Banach spaces, Israel Journal of Mathematics, 4, 213-216 (1966) · Zbl 0143.34904 · doi:10.1007/BF02771633
[2] Benyamini, Y.; Lindenstrauss, J., Geometric Nonlinear Functional Analysis Vol. 1 (2000), Providence: American Mathematical Society, Providence · Zbl 0946.46002
[3] Borwein, J. M.; Fitzpatrick, S., Existence of nearest points in Banach spaces, Canadian Journal of Mathematics, 41, 702-720 (1989) · Zbl 0668.46006
[4] Christensen, J. P. R., Measure theoretic zero sets in infinite dimensional spaces and applications to differentiability of Lipschitz mappings, Publications du Département de Mathématiques (Lyon), 10, 29-39 (1973) · Zbl 0302.43001
[5] Deville, R.; Godefroy, G.; Zizler, V., Smoothness and renormings in Banach spaces (1993), Harlow: Longman, Harlow · Zbl 0782.46019
[6] Duda, J.; Veselý, L.; Zajíček, L., On d.c. functions and mappings, Atti del Seminario Matematico e Fisico dell’Università di Modena, 51, 111-138 (2003) · Zbl 1072.46025
[7] Fitzpatrick, S., Differentiation of real-valued functions and continuity of metric projections, Proceedings of the American Mathematical Society, 91, 544-548 (1984) · Zbl 0604.46050 · doi:10.2307/2044798
[8] Fitzpatrick, S., Metric projections and the differentiability distance functions, Bulletin of the Australian Mathematical Society, 22, 291-312 (1980) · Zbl 0437.46012 · doi:10.1017/S0004972700006596
[9] Lau, K. S., Farthest points in weakly compact sets, Israel Journal of Mathematics, 22, 168-174 (1975) · Zbl 0325.46022 · doi:10.1007/BF02760164
[10] Matoušek, J.; Matoušková, E., A highly non-smooth norm on Hilbert space, Israel Journal of Mathematics, 112, 1-27 (1999) · Zbl 0935.46012 · doi:10.1007/BF02773476
[11] Matoušková, E., Almost nowhere Fréchet smooth norms, Studia Mathematica, 133, 93-99 (1999) · Zbl 0923.46016
[12] D. Preiss and L. Zajíček,Stronger estimates of sets of Frechet nondifferentiability of convex functions, Supplement to Rendiconti del Circolo Matematico di Palermo, Serie II No. 3 (1984), 219-223. · Zbl 0547.46026
[13] Zajíček, L., Differentiability of the distance function and points of multivaluedness of the metric projection in Banach space, Czechoslovak Mathematical Journal, 33, 108, 292-308 (1983) · Zbl 0527.41028
[14] Zajíček, L., Supergeneric results and Gateâux differentiability of convex Lipschitz functions on small sets, Acta Universitatis Carolinae, 38, 19-37 (1997)
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