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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
MathSciNet review: 1363408
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Abstract: Let $\mathbf {E}$ be a separable strictly convex Banach space of dimension at least 2. It is shown that there exists a nonempty compact connected set $X \subset % \mathbf {E}$ such that the nearest point mapping $p_X:% \mathbf {E}\to 2^{% \mathbf {E}}$ is not single valued on a set of points dense in $\mathbf {E}$. Furthermore, it is proved that most (in the sense of the Baire category) nonempty compact connected sets $X\subset % \mathbf {E}$ have the above property. Similar results hold for the furthest point mapping.


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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: http://dx.doi.org/10.1090/S0002-9939-96-03689-1
PII: S 0002-9939(96)03689-1
Received by editor(s): April 21, 1992
Communicated by: Dale E. Alspach
Article copyright: © Copyright 1996 American Mathematical Society