Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. M. Baronti and P. L. Papini, Diameters, centers and diametrically maximal sets, Rend. Circolo Mat. Palermo Suppl. (II) 38 (1995), 11-24. MR 1346789 (96f:52009)
  • 2. E. Behrends, and P. Harmand, Banach spaces which are proper M-ideals, Studia Math. 81 (1985), 159-169. MR 818178 (87f:46031)
  • 3. E. Behrends, Points of symmetry of convex sets in the two-dimensional complex space--a counterexample to D. Yost's problem. Math. Ann. 290 (1991), no. 3, 463-471. MR 1116232 (92f:46007)
  • 4. V. G. Boltyanski, H. Martini and P. S. Soltan, Excursions into Combinatorial Geometry, Springer, Berlin, 1997. MR 1439963 (98b:52001)
  • 5. G. D. Chakerian and H. Groemer, Convex bodies of constant width, in Convexity and its Applications, P. Gruber and J. Wills, Eds., Birkhäuser, 1983, 49-96. MR 731106 (85f:52001)
  • 6. H. G. Eggleston, Sets of constant width in finite-dimensional Banach spaces, Israel J. Math. 3 (1965), 163-172. MR 0200695 (34:583)
  • 7. L. Euler, De Curvis Triangularibus, Acta Academiae Scientarum Imperialis Petropolitinae (1778), 3-30. Opera Omnia: Series 1, Volume 28, pp. 298-321.
  • 8. C. Franchetti, Relationship between the Jung constant and a certain projection constant in Banach spaces, Ann. Univ. Ferrara, N. Ser. 23 (1977), 39-44. MR 0493263 (58:12292)
  • 9. H. Groemer, On complete convex bodies, Geom. Dedicata 20 (1986), 319-334. MR 845426 (87h:52008)
  • 10. E. Heil and H. Martini, Special convex bodies, in Handbook of Convex Geometry, Vol. A, P. Gruber and J. Wills, Eds., North-Holland, 1993, 347-385. MR 1242985 (94h:52001)
  • 11. T. Lachand-Robert and E. Oudet, Bodies of constant width in arbitrary dimensions, Math. Nach. 280 (7) (2007), 740-750. MR 2321138 (2008d:52002)
  • 12. E. Meissner, Über Punktmengen konstanter Breite, Vierteljahresschr. naturforsch. Ges Zürich 56 (1911), 42-50. Jbuch. 42, p. 91.
  • 13. J. P. Moreno, Semicontinuous functions and convex sets in $ C(K)$ spaces, J. Austral. Math. Soc. 82 (2007), 111-121. MR 2301973 (2008m:46036)
  • 14. J. P. Moreno, Porosity and diametrically maximal sets in $ C(K)$, Monatsh. Math. 152 (2007), 255-263. MR 2357520 (2008k:46050)
  • 15. J. P. Moreno, P. L. Papini and R. R. Phelps, Diametrically maximal and constant width sets in Banach spaces, Canad. J. Math. 58 (4) (2006), 820-842. MR 2245275 (2007j:52003)
  • 16. J. P. Moreno, P. L. Papini and R. R. Phelps, New families of convex sets related to diametral maximality, J. Convex. Anal. 13 (2006), 823-837. MR 2291568 (2008c:52001)
  • 17. J. P. Moreno and R. Schneider, Continuity properties of the ball hull mapping, Nonlinear Anal. 66 (2007), 914-925. MR 2288440 (2007m:46026)
  • 18. R. Paya and A. Rodríguez-Palacios, Banach spaces which are semi-L-summands in their biduals, Math. Ann. 289 (1991), 529-542. MR 1096186 (92e:46024)
  • 19. A. Rodríguez-Palacios, Infinite-dimensional sets of constant width and their applications, Extracta Math. Actas del II Congreso de Análisis Funcional, Jarandilla, España (1990), 140-151.
  • 20. A. Rodríguez-Palacios, Properly semi-L-embedded complex spaces, Studia Math. 106 (1993), 197-202. MR 1240314 (94k:46025)
  • 21. P. R. Scott, Sets of constant width and inequalities, Quart. J. Math. Oxford Ser. (2) 32 (1981), no. 127, 345-348. MR 625646 (82k:52013)
  • 22. S. Vrećica, A note on sets of constant width, Publ. Inst. Math. (Beograd) (N.S.) 29 (43) (1981), 289-291. MR 657118 (83g:52006)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 46E15, 52A05

Retrieve articles in all journals with MSC (2010): 46E15, 52A05

Additional Information

J. P. Moreno
Affiliation: Departamento Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Madrid 28049, Spain

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.

American Mathematical Society