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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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 $ \mathcal H$ be the family of all closed, convex and bounded subsets of a Banach space endowed with the Hausdorff metric. A completion of $ A\in\mathcal H$ is a diametrically maximal set $ D\in\mathcal H$ satisfying $ A\subset D$ and $ \operatorname{diam} A=\operatorname{diam} D$. The complete hull mapping associates with every $ A\in\mathcal H$ the family $ \gamma(A)$ of all its possible completions. It is shown that the set-valued mapping $ \gamma$ need not be convex valued even in finite-dimensional spaces, while, in the case of $ C(K)$ spaces, $ \gamma$ is convex valued if and only if $ K$ is extremally disconnected. Regarding the continuity we prove that, again in $ C(K)$ spaces, $ \gamma$ 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 $ \gamma$ 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: http://dx.doi.org/10.1090/S0002-9947-10-05142-1
PII: S 0002-9947(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.