Convex values and Lipschitz behavior of the complete hull mapping

Author:
J. P. Moreno

Journal:
Trans. Amer. Math. Soc. **362** (2010), 3377-3389

MSC (2010):
Primary 46E15, 52A05

DOI:
https://doi.org/10.1090/S0002-9947-10-05142-1

Published electronically:
February 24, 2010

MathSciNet review:
2601594

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Abstract: This note continues the study initiated in 2006 by P.L. Papini, R. R. Phelps and the author on some classical notions from finite-dimensional convex geometry in spaces of continuous functions. Let be the family of all closed, convex and bounded subsets of a Banach space endowed with the Hausdorff metric. A *completion* of is a diametrically maximal set satisfying and . The complete hull mapping associates with every the family of all its possible completions. It is shown that the set-valued mapping need not be convex valued even in finite-dimensional spaces, while, in the case of spaces, is convex valued if and only if is extremally disconnected. Regarding the continuity we prove that, again in spaces, is always Lipschitz continuous with constant less than or equal to 5 and has a Lipschitz selection with constant less than or equal to 3. If we consider the analogous problem in Euclidean spaces, we show that is Hölder continuous of order 1/4 and locally Hölder continuous of order 1/2, the Hölder constants depending on the diameter of the sets in both cases.

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Additional Information

**J. P. Moreno**

Affiliation:
Departamento Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Madrid 28049, Spain

Email:
josepedro.moreno@uam.es

DOI:
https://doi.org/10.1090/S0002-9947-10-05142-1

Keywords:
Diametrically maximal set,
constant width set,
complete hull mapping,
$C(K)$ space

Received by editor(s):
July 31, 2007

Published electronically:
February 24, 2010

Additional Notes:
This work was partially supported by the DGICYT project MTM 2006-03531

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.