Some remarks on the space of differences of sublinear functions

Sven Bartels; Diethard Pallaschke

Applicationes Mathematicae (1994)

  • Volume: 22, Issue: 3, page 419-426
  • ISSN: 1233-7234

Abstract

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Two properties concerning the space of differences of sublinear functions D(X) for a real Banach space X are proved. First, we show that for a real separable Banach space (X,‖·‖) there exists a countable family of seminorms such that D(X) becomes a Fréchet space. For X = ℝ^n this construction yields a norm such that D(ℝ^n) becomes a Banach space. Furthermore, we show that for a real Banach space with a smooth dual every sublinear Lipschitzian function can be expressed by the Fenchel conjugate of the farthest point mapping to its subdifferential at the origin. This leads to a simple family of sublinear functions which contains an exhaustive family of upper convex approximations for any quasidifferentiable function.

How to cite

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Bartels, Sven, and Pallaschke, Diethard. "Some remarks on the space of differences of sublinear functions." Applicationes Mathematicae 22.3 (1994): 419-426. <http://eudml.org/doc/219104>.

@article{Bartels1994,
abstract = {Two properties concerning the space of differences of sublinear functions D(X) for a real Banach space X are proved. First, we show that for a real separable Banach space (X,‖·‖) there exists a countable family of seminorms such that D(X) becomes a Fréchet space. For X = ℝ^n this construction yields a norm such that D(ℝ^n) becomes a Banach space. Furthermore, we show that for a real Banach space with a smooth dual every sublinear Lipschitzian function can be expressed by the Fenchel conjugate of the farthest point mapping to its subdifferential at the origin. This leads to a simple family of sublinear functions which contains an exhaustive family of upper convex approximations for any quasidifferentiable function.},
author = {Bartels, Sven, Pallaschke, Diethard},
journal = {Applicationes Mathematicae},
keywords = {upper convex approximation; sublinear function; Fenchel conjugation; quasidifferentiable function; differences; real-valued sublinear functions; directional derivative},
language = {eng},
number = {3},
pages = {419-426},
title = {Some remarks on the space of differences of sublinear functions},
url = {http://eudml.org/doc/219104},
volume = {22},
year = {1994},
}

TY - JOUR
AU - Bartels, Sven
AU - Pallaschke, Diethard
TI - Some remarks on the space of differences of sublinear functions
JO - Applicationes Mathematicae
PY - 1994
VL - 22
IS - 3
SP - 419
EP - 426
AB - Two properties concerning the space of differences of sublinear functions D(X) for a real Banach space X are proved. First, we show that for a real separable Banach space (X,‖·‖) there exists a countable family of seminorms such that D(X) becomes a Fréchet space. For X = ℝ^n this construction yields a norm such that D(ℝ^n) becomes a Banach space. Furthermore, we show that for a real Banach space with a smooth dual every sublinear Lipschitzian function can be expressed by the Fenchel conjugate of the farthest point mapping to its subdifferential at the origin. This leads to a simple family of sublinear functions which contains an exhaustive family of upper convex approximations for any quasidifferentiable function.
LA - eng
KW - upper convex approximation; sublinear function; Fenchel conjugation; quasidifferentiable function; differences; real-valued sublinear functions; directional derivative
UR - http://eudml.org/doc/219104
ER -

References

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  7. [7] D. Pallaschke, S. Scholtes and R. Urbański, On minimal pairs of compact convex sets, Bull. Polish Acad. Sci. Math. 39 (1991), 1-5. Zbl0759.52003
  8. [8] D. Pallaschke and R. Urbański, Some criteria for the minimality of pairs of compact convex sets, Z. Oper. Res. 37 (1993), 129-150. Zbl0781.49011
  9. [9] D. Pallaschke and R. Urbański, Reduction of quasidifferentials and minimal representations, Math. Programming Ser. A, to appear. Zbl0824.49016
  10. [10] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N.J., 1972. Zbl0224.49003
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  12. [12] H. H. Schäfer, Topological Vector Spaces, Springer, New York, 1971. 
  13. [13] S. Scholtes, Minimal pairs of convex bodies in two dimensions, Mathematika 39 (1992), 267-273. Zbl0759.52004

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