# 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

## Access Full Article

top## Abstract

top## How to cite

topBartels, 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

top- [1] U. Cegrell, On the space of delta-convex functions and its dual, Bull. Math. Soc. Sci. Math. R. S. Roumanie 22 (1978), 133-139.
- [2] V. F. Demyanov and A. M. Rubinov, Quasidifferential Calculus, Optimization Software Inc., Publications Division, New York, 1986.
- [3] J. Diestel, Geometry of Banach Spaces-Selected Topics, Lecture Notes in Math. 485, Springer, Heidelberg, 1975. Zbl0307.46009
- [4] J. Grzybowski, Minimal pairs of compact convex sets, Arch. Math. (Basel), submitted. Zbl0804.52002
- [5] L. Hörmander, Sur la fonction d'appui des ensembles convexes dans un espace localement convexe, Ark. Mat. 3 (1954), 181-186. Zbl0064.10504
- [6] D. Pallaschke, P. Recht and R. Urbański, On locally Lipschitz quasidifferentiable functions in Banach spaces, Optimization 17 (1986), 287-295. Zbl0597.49011
- [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] 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] D. Pallaschke and R. Urbański, Reduction of quasidifferentials and minimal representations, Math. Programming Ser. A, to appear. Zbl0824.49016
- [10] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N.J., 1972. Zbl0224.49003
- [11] S. Rolewicz, Metric Linear Spaces, PWN, Warszawa, and Reidel, Boston, 1984.
- [12] H. H. Schäfer, Topological Vector Spaces, Springer, New York, 1971.
- [13] S. Scholtes, Minimal pairs of convex bodies in two dimensions, Mathematika 39 (1992), 267-273. Zbl0759.52004

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.