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CHEBYSHEV SETS

Part of: Functional analysis Miscellaneous applications of operator theory Normed linear spaces and Banach spaces; Banach lattices Approximations and expansions Nonlinear operators and their properties

Published online by Cambridge University Press:  11 November 2014

JAMES FLETCHER  and
WARREN B. MOORS
JAMES FLETCHER
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong 2522, Australia email [email protected]
WARREN B. MOORS*
Affiliation:
Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand email [email protected]
*
✉[email protected]
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Abstract

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A Chebyshev set is a subset of a normed linear space that admits unique best approximations. In the first part of this paper we present some basic results concerning Chebyshev sets. In particular, we investigate properties of the metric projection map, sufficient conditions for a subset of a normed linear space to be a Chebyshev set, and sufficient conditions for a Chebyshev set to be convex. In the second half of the paper we present a construction of a nonconvex Chebyshev subset of an inner product space.

Keywords

convex setsChebyshev setsBanach spacesbest approximationsmetric projectionsuniquely remotal sets

MSC classification

Primary: 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
Secondary: 41A50: Best approximation, Chebyshev systems 46B20: Geometry and structure of normed linear spaces 47N10: Applications in optimization, convex analysis, mathematical programming, economics 47H05: Monotone operators and generalizations 46-02: Research exposition (monographs, survey articles)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 98 , Issue 2 , April 2015 , pp. 161 - 231
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Amir, D. and Deutsch, F., ‘Suns, moons and quasi-polyhedra’, J. Approx. Theory 6 (1972), 176–201.Google Scholar
Aronszajn, N., Introduction to the Theory of Hilbert Spaces (Research Foundation of Oklahoma A and M College, Stillwater, OK, 1950).Google Scholar
Ascoli, G., ‘Sugli spazi lineari metrici e le loro varietá lineari’, Ann. Mat. Pura Appl. (4) 10 (1932), 38–81; 203–232.CrossRefGoogle Scholar
Asplund, E., ‘Čebyšev sets in Hilbert spaces’, Trans. Amer. Math. Soc. 144 (1969), 235–240.Google Scholar
Assadi, A., Haghshenas, H. and Narang, T. D., ‘A look at proximinal and Chebyshev sets in Banach spaces’, Matematiche (Catania) 69 (2014), 71–87.Google Scholar
Balaganskii, V. S., ‘Approximative properties of sets in Hilbert space’, Mat. Zametki 31 (1982), 397–404.Google Scholar
Balaganskiĭ, V. S. and Vlasov, L. P., ‘The problem of the convexity of Chebyshev sets’, Russian Math. Surveys 51(6) (1996), 1127–1190.Google Scholar
Banach, S., Théorie des opérations linéaires (Monografje Matematyczne, Warsaw, 1932).Google Scholar
Baronti, M. and Papini, P. L., ‘Remotal sets revisted’, Taiwanese J. Math. 5 (2001), 367–373.CrossRefGoogle Scholar
Blatter, J., ‘Weiteste Punkte und nächste Punkte’, Rev. Roumaine Math. Pures Appl. 14 (1969), 615–621.Google Scholar
Borwein, J. M., ‘Proximality and Chebyshev sets’, Optim. Lett. 1 (2007), 21–32.Google Scholar
Borwein, J. M. and Fitzpatrick, S., ‘Existence of nearest points in Banach spaces’, Canad. J. Math. 41 (1989), 702–720.CrossRefGoogle Scholar
Borwein, J. M. and Preiss, D., ‘A smooth variational principle with applications to subdifferentiability and differentiability of convex functions’, Trans. Amer. Math. Soc. 303 (1987), 517–527.Google Scholar
Bourbaki, N., ‘Sur les espaces de Banach’, C. R. Acad. Sci. Paris 206 (1938), 1701–1704.Google Scholar
Bourgain, J., ‘ℓ ∞∕c 0has no equivalent strictly convex norm’, Proc. Amer. Math. Soc. 78 (1980), 225–226.Google Scholar
Brøndsted, A. and Rockafellar, R. T., ‘On the subdifferentiability of convex functions’, Proc. Amer. Math. Soc. 16 (1965), 605–611.Google Scholar
Brouwer, L., ‘Über Abbildung von Mannigfaltigkeiten’, Math. Ann. 71 (1911), 97–115.Google Scholar
Brown, A. L., ‘A rotund and reflexive space having a subspace of codimension two with a discontinuous metric projection’, Michigan Math. J. 21 (1974), 145–151.Google Scholar
Bunt, L., ‘Bijdrage tot de theorie der konvekse puntverzamelingen’, Thesis, University of Groningen, Amsterdam, 1934.Google Scholar
Busemann, H., ‘Note on a theorem on convex sets’, Mat. Tidsskrift B (1947), 32–34.Google Scholar
Busemann, H., The Geometry of Geodesics (Academic Press, New York, 1955).Google Scholar
Clarkson, J. A., ‘Uniformly convex spaces’, Trans. Amer. Math. Soc. 40(3) (1936), 396–414.CrossRefGoogle Scholar
Deutsch, F., ‘Existence of best approximations’, J. Approx. Theory 28 (1980), 132–154.CrossRefGoogle Scholar
Deutsch, F., ‘The convexity of Chebyshev sets in Hilbert space’, in: Topic in Polynomials of One and Several Variables and Their Applications (World Scientific, River Edge, NJ, 1993), 143–150.Google Scholar
Deutsch, F., Best Approximation in Inner Product Spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC 7 (Springer-Verlag, New York, 2001).Google Scholar
Diestel, J., Sequences and Series in Banach Spaces (Springer, New York, 1984).CrossRefGoogle Scholar
Duda, J., ‘On the size of the set of points where the metric projection is discontinuous’, J. Nonlinear Convex Anal. 7 (2006), 67–70.Google Scholar
Dunford, N. and Schwartz, J. T., Linear Operators I. General Theory (Interscience Publishers, London, 1958).Google Scholar
Dunford, N. and Schwartz, J. T., Linear Operators. Part II: Spectral Theory. Self Adjoint Operators in Hilbert Space (Interscience Publishers, New York, 1963).Google Scholar
Eberlein, W. F., ‘Weak compactness in Banach spaces I’, Proc. Nat. Acad. Sci. USA 33 (1947), 51–53.Google Scholar
Efimov, N. V. and Stečkin, S. B., ‘Some properties of Čebyšev sets’, Dokl. Akad. Nauk SSSR 118 (1958), 17–19; (in Russian).Google Scholar
Efinov, N. V. and Stečkin, S. B., ‘Čebyšev sets in Banach space’, Dokl. Akad. Nauk SSSR 121 (1958), 582–585; (in Russian).Google Scholar
Efimov, N. V. and Stečkin, S. B., ‘Support properties of sets in Banach spaces and Čebyšev sets’, Dokl. Akad. Nauk SSSR 127 (1959), 254–257.Google Scholar
Efimov, N. V. and Stečkin, S. B., ‘Approximative compactness and Chebyshev sets’, Dokl. Akad. Nauk SSSR 140 (1961), 522–524; (in Russian).Google Scholar
Ekeland, I., ‘Nonconvex minimization problems’, Bull. Amer. Math. Soc. (N.S.) 1(3) (1979), 443–474.Google Scholar
Fabian, M., Habala, P., Hájek, P., Montesinos, V. and Zizler, V., Banach Space Theory. The Basis for Linear and Nonlinear Analysis, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC (Springer, New York, 2011).CrossRefGoogle Scholar
Fitzpatrick, S., ‘Metric projections and the differentiability of distance functions’, Bull. Aust. Math. Soc. 22 (1980), 291–312.Google Scholar
Fréchet, M., ‘Sur les opérations linéaires III’, Trans. Amer. Math. Soc. 8 (1907), 433–446.CrossRefGoogle Scholar
Friedrichs, K. O., ‘On Clarkson’s inequalities’, Comm. Pure Appl. Math. 23 (1970), 603–607.CrossRefGoogle Scholar
Giles, J. R., Convex Analysis with Application in the Differentiation of Convex Functions, Research Notes in Mathematics, 58 (Pitman (Advanced Publishing Program), Boston, 1982).Google Scholar
Giles, J. R., ‘Differentiability of distance functions and a proximinal property inducing convexity’, Proc. Amer. Math. Soc. 104 (1988), 458–464.Google Scholar
Hu, Z., Moors, W. B. and Smith, M. A., ‘On a Banach space without a weak mid-point locally uniformly rotund norm’, Bull. Aust. Math. Soc. 56 (1997), 193–196.Google Scholar
James, R. C., ‘Weakly compact sets’, Trans. Amer. Math. Soc. 113 (1964), 129–140.Google Scholar
Jessen, B., ‘To sætninger om konvekse punktmængder’, Mat. Tidsskrift B (1940), 66–70.Google Scholar
Jiang, M., ‘On Johnson’s example of a nonconvex Chebyshev set’, J. Approx. Theory 74(2) (1993), 152–158.CrossRefGoogle Scholar
Johnson, G. G., ‘A nonconvex set which has the unique nearest point property’, J. Approx. Theory 51(4) (1987), 289–332.CrossRefGoogle Scholar
Kadec, M. I., ‘On strong and weak convergence’, Dokl. Akad. Nauk SSSR 122 (1958), 12–16.Google Scholar
Klee, V., ‘Convex bodies and periodic homeomorphisms in Hilbert spaces’, Trans. Amer. Math. Soc. 74 (1953), 10–43.Google Scholar
Klee, V. L., ‘Mappings into normed linear spaces’, Fund. Math. 49 (1960/61), 25–34.CrossRefGoogle Scholar
Klee, V. L., ‘Convexity of Chebyshev sets’, Math. Ann. 142 (1961), 292–304.Google Scholar
Klee, V., ‘Remarks on nearest points in normed linear spaces’, in: Proc. Colloquium on Convexity Copenhagen, 1965, Kobenhavns Univ. Mat. Inst., Copenhagen (1967), 168–176.Google Scholar
Kritikos, M., ‘Sur quelques propriétés des ensembles convexes’, Bull. Math. Soc. Romnine Sci. 40 (1938), 87–92.Google Scholar
Lindenstrauss, J., Preiss, D. and Tiser, J., Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces, Annals of Mathematics Studies, 179 (Princeton University Press, Princeton, NJ, 2012).Google Scholar
Mazur, S., ‘Über die kleinste konvexe Menge, die eine gegebene kompakte Menge enthält’, Stud. Math. 2 (1930), 7–9.Google Scholar
Moors, W. B., ‘An elementary proof of James’ characterisation of weak compactness’, Bull. Aust. Math. Soc. 84 (2011), 98–102.Google Scholar
Moors, W. B. and Sciffer, S. D., ‘Sigma-fragmentable spaces that are not a countable union of fragmentable spaces’, Topol. Appl. 119 (2002), 279–286.CrossRefGoogle Scholar
Motzkin, T., ‘Sur quelques propriétés caractéristiques des ensembles convexes’, Rend. Acad. dei Lincei (Roma) 21 (1935), 562–567.Google Scholar
Motzkin, T., ‘Sur quelques propriétés caractéristiques des ensembles bornés non convexes’, Rend. Acad. dei Lincei (Roma) 21(series 6) (1935), 773–779.Google Scholar
Narang, T. D., ‘Convexity of Chebyshev sets’, Nieuw Arch. Wiskd. (5) 25 (1977), 377–402.Google Scholar
Narang, T. D., ‘Uniquely remotal sets are singletons’, Nieuw Arch. Wiskd. (5) 9(4) (1991), 1–12.Google Scholar
Phelps, R. R., ‘Convex sets and nearest points’, Proc. Amer. Math. Soc. 8 (1957), 790–797.Google Scholar
Phelps, R. R., Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics, 1364 (Springer, Berlin, 1993).Google Scholar
Preiss, D., ‘Differentiability of Lipschitz functions on Banach spaces’, J. Funct. Anal. 91 (1990), 312–345.Google Scholar
Rademacher, H., ‘Über partielle und totale Differenzierbarkeit von Funktionen mehrerer Variabeln und über die Transformation der Doppelintegrale’, Math. Ann. 79 (1919), 340–359.Google Scholar
Radon, J., ‘Theorie und Anwendugen der absolut additiven Mengenfunctionen’, Sitz. Akad. Wiss. Wien. 122 (1913), 1295–1438.Google Scholar
Rao, M. and Stetkær, H., Complex Analysis. An Invitation. A Concise Introduction to Complex Function Theory (World Scientific, Teaneck, NJ, 1991).Google Scholar
Riesz, F., ‘Sur une espèce de géométrie analytique des systémes de fonctions sommables’, C. R. Acad. Sci. Paris 144 (1907), 1409–1411.Google Scholar
Riesz, F., ‘Untersuchungen über Systeme integrierbarer Funktionen’, Math. Ann. 69 (1910), 449–497.Google Scholar
Riesz, F., ‘Sur la convergence en moyenne I’, Acta Sci. Math. 4 (1928/29), 58–64.Google Scholar
Riesz, F., ‘Sur la convergence en moyenne II’, Acta Sci. Math. 4 (1928/29), 182–185.Google Scholar
Revalski, J. P. and Zhivkov, N. V., ‘Best approximation problems in compactly uniformly rotund spaces’, J. Convex Anal. 19 (2012), 1153–1166.Google Scholar
Rockafellar, R. T., ‘On the virtual convexity of the domain and range of a nonlinear maximal monotone operator’, Math. Ann. 185 (1970), 81–90.Google Scholar
Rockafellar, R. T., ‘On the maximal monotonicity of subdifferential mappings’, Pacific J. Math. 33 (1970), 209–216.CrossRefGoogle Scholar
Royden, H. L., Real Analysis, 3rd edn (Macmillan, New York, 1988).Google Scholar
Rudin, W., Functional Analysis, 2nd edn, International Series in Pure and Applied Mathematics (McGraw-Hill, New York, 1991).Google Scholar
Raymond, J. S., ‘Weak compactness and variational characterisation of the convexity’, Mediterr. J. Math. 10 (2013), 927–940.Google Scholar
Schauder, J., ‘Der Fixpunktsatz in Funktionalräumen’, Stud. Math. 2(1) (1930), 171–180.Google Scholar
Singer, I., Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces (Springer, New York, 1970).Google Scholar
Singer, I., The Theory of Best Approximation in Functional Analysis, Conference Board of the Mathematical Sciences Regional Series in Applied Mathematics, 13 (Society for Industrial and Applied Mathematics, Philadelphia, 1974).Google Scholar
Šmulian, V. L., ‘Sur la dérivabilité de la norme dans l’espace de Banach’, C. R. Acad. Sci. URSS (Dokl.) N.S. 27 (1940), 643–648.Google Scholar
Stečkin, S. B., ‘Approximation properties of sets in normed linear spaces’, Rev. Math. Pures Appl. 8 (1963), 5–18.Google Scholar
Troyanski, S., ‘On locally uniformly convex and differentiable norms in certain nonseparable Banach spaces’, Stud. Math. 37 (1971), 173–180.CrossRefGoogle Scholar
Valentine, F. A., Convex Sets, McGraw-Hill Series in Higher Mathematics (McGraw-Hill, New York–Toronto–London, 1964).Google Scholar
Vlasov, L. P., ‘Chebyshev sets in Banach spaces’, Dokl. Akad. Nauk SSSR 141 (1961), 19–20.Google Scholar
Vlasov, L. P., ‘On Čebyšev sets’, Dokl. Akad. Nauk SSSR 173 (1967), 491–494.Google Scholar
Vlasov, L. P., ‘Almost convex and Chebyshev sets’, Math. Notes Acad. Sci. USSR 8 (1970), 776–779.Google Scholar
Vlasov, L. P., ‘Approximative properties of sets in normed linear spaces’, Russian Math. Surveys 28 (1973), 1–95.Google Scholar
Westphal, U. and Frerking, J., ‘On a property of metric projections onto closed subsets of Hilbert spaces’, Proc. Amer. Math. Soc. 105 (1989), 644–651.Google Scholar
Wu, Z., ‘A Chebyshev set and its distance function’, J. Approx. Theory 119 (2002), 181–192.Google Scholar