On the Infimum of a Quasiconvex Function over an Intersection. Application to the Distance Function

Juan-Enrique Martínez-Legaz; Antonio Martinón

doi:10.68381/jca18043

Abstract

We give sufficient conditions for the infimum of a quasiconvex function

f

over the intersection

\bigcap_{i\in I}R_{i}

to agree with the supremum of the infima of

f

over the

R_{i}

's. We apply these results to the distance function in a normed space.

Author information

Contact details are reproduced from the original publication and may be historical.

Juan-Enrique Martínez-Legaz

Dep. d'Economia i d'Història Econòmica, Universitat Autònoma de Barcelona, 08193 Bellaterra - Barcelona, Spain
and: MOVE (Markets, Organizations and Votes in Economics)

JuanEnrique.Martinez.Legaz@uab.es

Antonio Martinón

Dep. de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna - Tenerife, Spain

anmarce@ull.es

Keywords

Quasiconvex functions
distance to the intersection

Mathematics Subject Classification

54E35; 26-99

References

J. M. F. Castillo, P. L. Papini: Approximation of the limit distance function in Banach spaces, J. Math. Anal. Appl. 328 (2007) 577–589.https://doi.org/10.1016/j.jmaa.2006.05.042 Google Scholar
G. Chacón, V. Montesinos, A. Octavio: A note on the intersection of Banach subspaces, Publ. Res. Inst. Math. Sci. 40 (2004) 1–6.https://doi.org/10.2977/prims/1145475964 Google Scholar
O. Dovgoshey, O. Martio: Blow up of balls and coverings in metric spaces, Manuscr. Math. 127 (2008) 89–120.https://doi.org/10.1007/s00229-008-0191-x Google Scholar
A. Hoffmann: The distance to the intersection of two convex sets expressed by the distance to each of them, Math. Nachr. 157 (1992) 81–98.https://doi.org/10.1002/mana.19921570108 Google Scholar
J. Lindenstrauss, L. Tzafriri: Classical Banach Spaces I, Springer, Berlin (1977).https://doi.org/10.1007/978-3-642-66557-8 Google Scholar
J.-E. Martínez-Legaz, A. Martinón: Boundedly connected sets and the distance to the intersection of two sets, J. Math. Anal. Appl. 332 (2007) 400–406.https://doi.org/10.1016/j.jmaa.2006.10.037 Google Scholar
J.-E. Martínez-Legaz, A. M. Rubinov, I. Singer: Downward sets and their separation and approximation properties, J. Glob. Optim. 23 (2002) 111–137.https://doi.org/10.1023/A:1015583411806 Google Scholar
A. Martinón: Distance to the intersection of two sets, Bull. Aust. Math. Soc. 70 (2004) 329–341.https://doi.org/10.1017/S0004972700034547 Google Scholar
A. M. Rubinov: Distance to the solution set of an inequality with an increasing function, in: Equilibrium Problems and Variational Models (Erice, 2000), P. Danilele et al. (ed.), Kluwer, Boston (2003) 417–431.https://doi.org/10.1007/978-1-4613-0239-1_23 Google Scholar
A. M. Rubinov, I. Singer: Best approximation by normal and conormal sets, J. Approx. Theory 107 (2000) 212–243.https://doi.org/10.1006/jath.2000.3495 Google Scholar
A. M. Rubinov, I. Singer: Distance to the intersection of normal sets and applications, Numer. Funct. Anal. Optim. 21 (2000) 521–535.https://doi.org/10.1080/01630560008816970 Google Scholar
S. Willard: General Topology, Addison-Wesley, Reading (1970).Google Scholar

Suggested citation

Published by Heldermann Verlag, 2011. Rights now held by Banach Press LLC.

[1] J. M. F. Castillo, P. L. Papini: Approximation of the limit distance function in Banach spaces, J. Math. Anal. Appl. 328 (2007) 577–589.https://doi.org/10.1016/j.jmaa.2006.05.042 Google Scholar

[2] G. Chacón, V. Montesinos, A. Octavio: A note on the intersection of Banach subspaces, Publ. Res. Inst. Math. Sci. 40 (2004) 1–6.https://doi.org/10.2977/prims/1145475964 Google Scholar

[3] O. Dovgoshey, O. Martio: Blow up of balls and coverings in metric spaces, Manuscr. Math. 127 (2008) 89–120.https://doi.org/10.1007/s00229-008-0191-x Google Scholar

[4] A. Hoffmann: The distance to the intersection of two convex sets expressed by the distance to each of them, Math. Nachr. 157 (1992) 81–98.https://doi.org/10.1002/mana.19921570108 Google Scholar

[5] J. Lindenstrauss, L. Tzafriri: Classical Banach Spaces I, Springer, Berlin (1977).https://doi.org/10.1007/978-3-642-66557-8 Google Scholar

[6] J.-E. Martínez-Legaz, A. Martinón: Boundedly connected sets and the distance to the intersection of two sets, J. Math. Anal. Appl. 332 (2007) 400–406.https://doi.org/10.1016/j.jmaa.2006.10.037 Google Scholar

[7] J.-E. Martínez-Legaz, A. M. Rubinov, I. Singer: Downward sets and their separation and approximation properties, J. Glob. Optim. 23 (2002) 111–137.https://doi.org/10.1023/A:1015583411806 Google Scholar

[8] A. Martinón: Distance to the intersection of two sets, Bull. Aust. Math. Soc. 70 (2004) 329–341.https://doi.org/10.1017/S0004972700034547 Google Scholar

[9] A. M. Rubinov: Distance to the solution set of an inequality with an increasing function, in: Equilibrium Problems and Variational Models (Erice, 2000), P. Danilele et al. (ed.), Kluwer, Boston (2003) 417–431.https://doi.org/10.1007/978-1-4613-0239-1_23 Google Scholar

[10] A. M. Rubinov, I. Singer: Best approximation by normal and conormal sets, J. Approx. Theory 107 (2000) 212–243.https://doi.org/10.1006/jath.2000.3495 Google Scholar

[11] A. M. Rubinov, I. Singer: Distance to the intersection of normal sets and applications, Numer. Funct. Anal. Optim. 21 (2000) 521–535.https://doi.org/10.1080/01630560008816970 Google Scholar

[12] S. Willard: General Topology, Addison-Wesley, Reading (1970).Google Scholar