### Abstract

In this paper, we investigate properties of set-valued mappings that establish connection between the values of this map at two arbitrary points of the domain and the value at their midpoint. Such properties are, for instance, Jensen convexity/concavity, K-Jensen convexity/concavity (where K is the set of nonnegative elements of an ordered vector space), and approximate/strong K-Jensen convexity/concavity. Assuming weak but natural regularity assumptions on the set-valued map, our main purpose is to deduce the convexity/concavity consequences of these properties in the appropriate sense. Our two main theorems will generalize most of the known results in this field, in particular the celebrated Bernstein - Doetsch Theorem from 1915, and thus they offer a unified view of these theories.

Original language | English |
---|---|

Pages (from-to) | 229-252 |

Number of pages | 24 |

Journal | Publicationes Mathematicae |

Volume | 84 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 2014 |

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### Keywords

- Approximate convexity
- K-Jensen convexity/concavity
- Set-valued map
- Strong convexity
- Takagi transformation

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Publicationes Mathematicae*,

*84*(1-2), 229-252. https://doi.org/10.5486/PMD.2014.5904

**Bernstein - Doetsch type theorems for set-valued maps of strongly and approximately convex and concave type.** / González, Carlos; Nikodem, Kazimierz; Páles, Z.; Roa, Gari.

Research output: Contribution to journal › Article

*Publicationes Mathematicae*, vol. 84, no. 1-2, pp. 229-252. https://doi.org/10.5486/PMD.2014.5904

}

TY - JOUR

T1 - Bernstein - Doetsch type theorems for set-valued maps of strongly and approximately convex and concave type

AU - González, Carlos

AU - Nikodem, Kazimierz

AU - Páles, Z.

AU - Roa, Gari

PY - 2014

Y1 - 2014

N2 - In this paper, we investigate properties of set-valued mappings that establish connection between the values of this map at two arbitrary points of the domain and the value at their midpoint. Such properties are, for instance, Jensen convexity/concavity, K-Jensen convexity/concavity (where K is the set of nonnegative elements of an ordered vector space), and approximate/strong K-Jensen convexity/concavity. Assuming weak but natural regularity assumptions on the set-valued map, our main purpose is to deduce the convexity/concavity consequences of these properties in the appropriate sense. Our two main theorems will generalize most of the known results in this field, in particular the celebrated Bernstein - Doetsch Theorem from 1915, and thus they offer a unified view of these theories.

AB - In this paper, we investigate properties of set-valued mappings that establish connection between the values of this map at two arbitrary points of the domain and the value at their midpoint. Such properties are, for instance, Jensen convexity/concavity, K-Jensen convexity/concavity (where K is the set of nonnegative elements of an ordered vector space), and approximate/strong K-Jensen convexity/concavity. Assuming weak but natural regularity assumptions on the set-valued map, our main purpose is to deduce the convexity/concavity consequences of these properties in the appropriate sense. Our two main theorems will generalize most of the known results in this field, in particular the celebrated Bernstein - Doetsch Theorem from 1915, and thus they offer a unified view of these theories.

KW - Approximate convexity

KW - K-Jensen convexity/concavity

KW - Set-valued map

KW - Strong convexity

KW - Takagi transformation

UR - http://www.scopus.com/inward/record.url?scp=84898678728&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84898678728&partnerID=8YFLogxK

U2 - 10.5486/PMD.2014.5904

DO - 10.5486/PMD.2014.5904

M3 - Article

AN - SCOPUS:84898678728

VL - 84

SP - 229

EP - 252

JO - Publicationes Mathematicae

JF - Publicationes Mathematicae

SN - 0033-3883

IS - 1-2

ER -