Hostname: page-component-f554764f5-68cz6 Total loading time: 0 Render date: 2025-04-10T02:02:48.972Z Has data issue: false hasContentIssue false
  • English
  • Français

Distance to the intersection of two sets

Published online by Cambridge University Press:  17 April 2009

Antonio Martinón
Antonio Martinón
Affiliation:
Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna (Tenerife), Spain e-mail: anmarce@ull.es
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give sufficient conditions so that the distance of a point to the intersection of two sets agrees with the maximum of the distances to each of them. The results are established in several settings: complete metric spaces, Banach spaces and spaces of subsets of Banach spaces.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 70 , Issue 2 , October 2004 , pp. 329 - 341
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Banaś, J. and Martinón, A., ‘Some properties of the Hausdorff distance in metric spaces’, Bull. Austral. Math. Soc. 42 (1990), 511–516.CrossRefGoogle Scholar
[2]Bantegnie, R., ‘Convexité des hyperespaces’, Arch. Math. 26 (1975), 414–420.CrossRefGoogle Scholar
[3]Blumenthal, L.M., Theory and applications of distance geometry (Clarendom Press, Oxford, 1953).Google Scholar
[4]Blumenthal, L.M. and Menger, K., Studies in Geometry (Freeman, San Francisco, 1970).Google Scholar
[5]Goebel, K. and Kirk, W.A., Topics in metric fixed point theory (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar
[6]Hoffmann, A., ‘The distance to the intersection of two convex sets expressed by the distance to each of them’, Math. Nachr. 157 (1992), 81–98.CrossRefGoogle Scholar
[7]Jongmans, F., ‘De l'art d'être à bonne distance des ensembles dont la décomposition atteint un stade avancé’, Bull. Soc. Roy. Sci. Liège 48 (1979), 237–261.Google Scholar
[8]Martínez-Legaz, J.E., Rubinov, A. M. and Singer, I., ‘Downward sets and their separation and approximation properties’, J. Global Optim. 23 (2002), 111–137.CrossRefGoogle Scholar
[9]Megginson, R., An introduction to Banach space theory (Springer-Verlag, New York, 1998).CrossRefGoogle Scholar
[10]Menger, K., ‘Untersuchungen über allgemeine Metrik’, Math. Ann. 100 (1928), 75–163.CrossRefGoogle Scholar
[11]Rubinov, A.M., ‘Distance to the solution set of and inequality with an increasing function’, in Equilibrium Problems and Variational Models, (Maugeri, A. and Danilele, P., Editors), Nonconvex Optim. Appl. 68 (Kluwer Acad. Pub., Norwell MA, 2003), pp. 417–431.CrossRefGoogle Scholar
[12]Rubinov, A.M. and Singer, I., ‘Best approximation by normal and conormal sets’, J. Approx. Theory 107 (2000), 212–243.CrossRefGoogle Scholar
[13]Rubinov, A.M. and Singer, I., ‘Distance to the intersection of normal sets and applications’, Numer. Funct. Anal. Optim. 21 (2000), 521–535.CrossRefGoogle Scholar