Strengthening of strong and approximate convexity

J. Makó, Z. Páles

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

Let X be a real linear space and D {subset double equals} X be a nonempty convex subset. Given an error function E:[0,1]×(D-D)→ℝ∪{+∞} and an element t ∈]0, 1[, a function f:D→ℝ is called (E,t)-convex if f(tx+(1-t)y)≦ tf(x)+(1-t)f(y)+E(t,x-y) for all x,y∈D. The main result of this paper states that, for all a,b∈(ℕ∪{0})+{0,t,1-t} such that {a,b,a+b}∩ℕ≠, every (E,t)-convex function is also, convex, where, As a consequence, under further assumptions on E, the strong and approximate convexity properties of (E,t)-convex functions can be strengthened.

Original languageEnglish
Pages (from-to)78-91
Number of pages14
JournalActa Mathematica Hungarica
Volume132
Issue number1-2
DOIs
Publication statusPublished - Jul 2011

Fingerprint

Strengthening
Convexity
Convex function
Subset
Error function
Linear Space

Keywords

  • (E,t)-convexity
  • 39B12
  • 39B22
  • approximate and strong convexity

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Strengthening of strong and approximate convexity. / Makó, J.; Páles, Z.

In: Acta Mathematica Hungarica, Vol. 132, No. 1-2, 07.2011, p. 78-91.

Research output: Contribution to journalArticle

@article{7285c98f5184401fba87e32d69226636,
title = "Strengthening of strong and approximate convexity",
abstract = "Let X be a real linear space and D {subset double equals} X be a nonempty convex subset. Given an error function E:[0,1]×(D-D)→ℝ∪{+∞} and an element t ∈]0, 1[, a function f:D→ℝ is called (E,t)-convex if f(tx+(1-t)y)≦ tf(x)+(1-t)f(y)+E(t,x-y) for all x,y∈D. The main result of this paper states that, for all a,b∈(ℕ∪{0})+{0,t,1-t} such that {a,b,a+b}∩ℕ≠, every (E,t)-convex function is also, convex, where, As a consequence, under further assumptions on E, the strong and approximate convexity properties of (E,t)-convex functions can be strengthened.",
keywords = "(E,t)-convexity, 39B12, 39B22, approximate and strong convexity",
author = "J. Mak{\'o} and Z. P{\'a}les",
year = "2011",
month = "7",
doi = "10.1007/s10474-010-0056-0",
language = "English",
volume = "132",
pages = "78--91",
journal = "Acta Mathematica Hungarica",
issn = "0236-5294",
publisher = "Springer Netherlands",
number = "1-2",

}

TY - JOUR

T1 - Strengthening of strong and approximate convexity

AU - Makó, J.

AU - Páles, Z.

PY - 2011/7

Y1 - 2011/7

N2 - Let X be a real linear space and D {subset double equals} X be a nonempty convex subset. Given an error function E:[0,1]×(D-D)→ℝ∪{+∞} and an element t ∈]0, 1[, a function f:D→ℝ is called (E,t)-convex if f(tx+(1-t)y)≦ tf(x)+(1-t)f(y)+E(t,x-y) for all x,y∈D. The main result of this paper states that, for all a,b∈(ℕ∪{0})+{0,t,1-t} such that {a,b,a+b}∩ℕ≠, every (E,t)-convex function is also, convex, where, As a consequence, under further assumptions on E, the strong and approximate convexity properties of (E,t)-convex functions can be strengthened.

AB - Let X be a real linear space and D {subset double equals} X be a nonempty convex subset. Given an error function E:[0,1]×(D-D)→ℝ∪{+∞} and an element t ∈]0, 1[, a function f:D→ℝ is called (E,t)-convex if f(tx+(1-t)y)≦ tf(x)+(1-t)f(y)+E(t,x-y) for all x,y∈D. The main result of this paper states that, for all a,b∈(ℕ∪{0})+{0,t,1-t} such that {a,b,a+b}∩ℕ≠, every (E,t)-convex function is also, convex, where, As a consequence, under further assumptions on E, the strong and approximate convexity properties of (E,t)-convex functions can be strengthened.

KW - (E,t)-convexity

KW - 39B12

KW - 39B22

KW - approximate and strong convexity

UR - http://www.scopus.com/inward/record.url?scp=79957909374&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79957909374&partnerID=8YFLogxK

U2 - 10.1007/s10474-010-0056-0

DO - 10.1007/s10474-010-0056-0

M3 - Article

VL - 132

SP - 78

EP - 91

JO - Acta Mathematica Hungarica

JF - Acta Mathematica Hungarica

SN - 0236-5294

IS - 1-2

ER -