On compact connected sets in Banach spaces

Authors:
F. S. de Blasi and J. Myjak

Journal:
Proc. Amer. Math. Soc. **124** (1996), 2331-2336

MSC (1991):
Primary 47A52; Secondary 46B20, 54E52

DOI:
https://doi.org/10.1090/S0002-9939-96-03689-1

MathSciNet review:
1363408

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a separable strictly convex Banach space of dimension at least 2. It is shown that there exists a nonempty compact connected set such that the nearest point mapping is not single valued on a set of points dense in . Furthermore, it is proved that most (in the sense of the Baire category) nonempty compact connected sets have the above property. Similar results hold for the furthest point mapping.

**1.**J. J. Schäffer,*Geometry of spheres in normed spaces*, Dekker, New York, 1976. MR**57:7120****2.**S. B. Steckin,*Approximation properties of sets in normed linear spaces*, Rev. Roumaine Math. Pures Appl.**8**(1963), pp. 5--13. MR**27:5018****3.**T. Zamfirescu,*The nearest point mapping is single valued nearly everywhere*, Arch. Math. (Basel)**51**(1990), pp. 563--566. MR**91k:41061**

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Additional Information

**F. S. de Blasi**

Affiliation:
Dipartimento di Matematica, Università di Roma II (Tor Vergata), Via della Ricerca Scientifica, 00133 Roma, Italy

**J. Myjak**

Affiliation:
Dipartimento di Matematica, Università di L’Aquila, Via Vetoio, 67100 L’Aquila, Italy

Email:
myjak@axscaq.aquila.imtn.it

DOI:
https://doi.org/10.1090/S0002-9939-96-03689-1

Received by editor(s):
April 21, 1992

Communicated by:
Dale E. Alspach

Article copyright:
© Copyright 1996
American Mathematical Society