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Mazur's intersection property of balls for compact convex sets

  • J. H. M. Whitfield (a1) and V. Zizler (a2)

Abstract

We show that every compact convex set in a Banach space X is an intersection of balls provided the cone generated by the set of all extreme points of the dual unit ball of X* is dense in X* in the topology of uniform convergence on compact sets in X. This allows us to renorm every Banach space with transfinite Schauder basis by a norm which shares the mentioned intersection property.

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References

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[1]Bessaga, C. and Pelczynski, A., Selected topics in infinite dimensional topology, Polish Sci. Publ. Warsaw 1975.
[2]Choquet, G., Lectures on Analysis, Vol. II, Representation Theory, Edited by Marsden, J., W.A. Benjamin, Inc. 1969.
[3]Giles, J.R., Gregory, D.A. and Sims, Brailey, “Characterization of normed linear spaces with Mazur's intersection property”, Bull. Austral. Math. Soc. 18 (1978), 105123.
[4]Mazur, S., “Uber Schwach Konvergenz in der Raumen (Lp)”, Studia. Math. 4 (1933), 128133.
[5]Phelps, R.R., “A representation theorem for bounded convex sets”, Proc. Amer. Math. Soc. 11 (1960), 976983.
[6]Rainwater, John, “Local uniform convexity of Day's norm on c0 (Γ)”, Proc. Amer. Math. Soc. 22 (1969), 335339.
[7]Rainwater, John, “Day's norm on c0 (Γ)”, Various Publ. Ser. No. 8. (1969).
[8]Sullivan, F., “Dentability, “Smoothability and stronger properties in Banach spaces”, Indiana Math. J. 26 (1977), 545553.
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Mazur's intersection property of balls for compact convex sets

  • J. H. M. Whitfield (a1) and V. Zizler (a2)

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