# Averages of uniformly continuous retractions

A. Jiménez-Vargas; J. Mena-Jurado; R. Nahum; J. Navarro-Pascual

Studia Mathematica (1999)

- Volume: 135, Issue: 1, page 75-81
- ISSN: 0039-3223

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topJiménez-Vargas, A., et al. "Averages of uniformly continuous retractions." Studia Mathematica 135.1 (1999): 75-81. <http://eudml.org/doc/216644>.

@article{Jiménez1999,

abstract = {Let X be an infinite-dimensional complex normed space, and let B and S be its closed unit ball and unit sphere, respectively. We prove that the identity map on B can be expressed as an average of three uniformly retractions of B onto S. Moreover, for every 0≤ r < 1, the three retractions are Lipschitz on rB. We also show that a stronger version where the retractions are required to be Lipschitz does not hold.},

author = {Jiménez-Vargas, A., Mena-Jurado, J., Nahum, R., Navarro-Pascual, J.},

journal = {Studia Mathematica},

keywords = {uniformly retraction; Lipschitz retraction; extreme point; uniformly continuous; averages; retractions; continuous retractions},

language = {eng},

number = {1},

pages = {75-81},

title = {Averages of uniformly continuous retractions},

url = {http://eudml.org/doc/216644},

volume = {135},

year = {1999},

}

TY - JOUR

AU - Jiménez-Vargas, A.

AU - Mena-Jurado, J.

AU - Nahum, R.

AU - Navarro-Pascual, J.

TI - Averages of uniformly continuous retractions

JO - Studia Mathematica

PY - 1999

VL - 135

IS - 1

SP - 75

EP - 81

AB - Let X be an infinite-dimensional complex normed space, and let B and S be its closed unit ball and unit sphere, respectively. We prove that the identity map on B can be expressed as an average of three uniformly retractions of B onto S. Moreover, for every 0≤ r < 1, the three retractions are Lipschitz on rB. We also show that a stronger version where the retractions are required to be Lipschitz does not hold.

LA - eng

KW - uniformly retraction; Lipschitz retraction; extreme point; uniformly continuous; averages; retractions; continuous retractions

UR - http://eudml.org/doc/216644

ER -

## References

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- [3] J. Cantwell, A topological approach to extreme points in function spaces, ibid. 19 (1968), 821-825. Zbl0175.13403
- [4] A. Jiménez-Vargas, J. F. Mena-Jurado and J. C. Navarro-Pascual, Complex extremal structure in spaces of continuous functions, J. Math. Anal. Appl. 211 (1997), 605-615. Zbl0920.46028
- [5] P. K. Lin and Y. Sternfeld, Convex sets with the Lipschitz fixed point property are compact, Proc. Amer. Math. Soc. 93 (1985) 633-639. Zbl0566.47039
- [6] J. C. Navarro-Pascual, Extreme points and retractions in Banach spaces, Israel J. Math. 99 (1997), 335-342. Zbl0901.46015
- [7] B. Nowak, On the Lipschitzian retraction of the unit ball in infinite-dimensional Banach spaces onto its boundary, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), 861-864. Zbl0472.54008
- [8] N. T. Peck, Extreme points and dimension theory, Pacific J. Math. 25 (1968), 341-351. Zbl0157.29603

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