# Polynomial selections and separation by polynomials

Studia Mathematica (1996)

- Volume: 120, Issue: 1, page 75-82
- ISSN: 0039-3223

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topWąsowicz, Szymon. "Polynomial selections and separation by polynomials." Studia Mathematica 120.1 (1996): 75-82. <http://eudml.org/doc/216322>.

@article{Wąsowicz1996,

abstract = {K. Nikodem and the present author proved in [3] a theorem concerning separation by affine functions. Our purpose is to generalize that result for polynomials. As a consequence we obtain two theorems on separation of an n-convex function from an n-concave function by a polynomial of degree at most n and a stability result of Hyers-Ulam type for polynomials.},

author = {Wąsowicz, Szymon},

journal = {Studia Mathematica},

keywords = {separation theorem; set-valued function; selection; n-convex function; n-concave function; affine function; Helly's theorem; Lagrange interpolating polynomial; set-valued functions; Lagrange interpolation polynomial; selections; separation by polynomials; affine functions; -convex function; -concave function},

language = {eng},

number = {1},

pages = {75-82},

title = {Polynomial selections and separation by polynomials},

url = {http://eudml.org/doc/216322},

volume = {120},

year = {1996},

}

TY - JOUR

AU - Wąsowicz, Szymon

TI - Polynomial selections and separation by polynomials

JO - Studia Mathematica

PY - 1996

VL - 120

IS - 1

SP - 75

EP - 82

AB - K. Nikodem and the present author proved in [3] a theorem concerning separation by affine functions. Our purpose is to generalize that result for polynomials. As a consequence we obtain two theorems on separation of an n-convex function from an n-concave function by a polynomial of degree at most n and a stability result of Hyers-Ulam type for polynomials.

LA - eng

KW - separation theorem; set-valued function; selection; n-convex function; n-concave function; affine function; Helly's theorem; Lagrange interpolating polynomial; set-valued functions; Lagrange interpolation polynomial; selections; separation by polynomials; affine functions; -convex function; -concave function

UR - http://eudml.org/doc/216322

ER -

## References

top- [1] E. Behrends and K. Nikodem, A selection theorem of Helly type and its applications, Studia Math. 116 (1995), 43-48. Zbl0847.52004
- [2] Z. Ciesielski, Some properties of convex functions of higher orders, Ann. Polon. Math. 7 (1959), 1-7. Zbl0128.05704
- [3] K. Nikodem and S. Wąsowicz, A sandwich theorem and Hyers-Ulam stability of affine functions, Aequationes Math. 49 (1995), 160-164. Zbl0815.39010
- [4] T. Popoviciu, Les Fonctions Convexes, Hermann, Paris, 1944. Zbl0060.14911
- [5] T. Popoviciu, Sur quelques propriétés des fonctions d'une variable réelle convexes d'ordre supérieur, Mathematica (Cluj) 8 (1934), 1-85. Zbl0009.05901
- [6] A. W. Roberts and D. E. Varberg, Convex Functions, Academic Press, New York, 1973. Zbl0271.26009
- [7] F. A. Valentine, Convex Sets, McGraw-Hill, New York, 1964.
- [8] S. Wąsowicz, On affine selections of set-valued functions, J. Appl. Anal. 1 (1995), 173-179.

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