Details
| Original language | English |
|---|---|
| Pages (from-to) | 1363-1375 |
| Number of pages | 13 |
| Journal | Journal of Symbolic Logic |
| Volume | 83 |
| Issue number | 4 |
| Early online date | 21 Dec 2018 |
| Publication status | Published - Dec 2018 |
| Externally published | Yes |
Abstract
In the framework of Bishop's constructive mathematics we introduce co-convexity as a property of subsets B of , the set of finite binary sequences, and prove that co-convex bars are uniform. Moreover, we establish a canonical correspondence between detachable subsets B of and uniformly continuous functions f defined on the unit interval such that B is a bar if and only if the corresponding function f is positive-valued, B is a uniform bar if and only if f has positive infimum, and B is co-convex if and only if f satisfies a weak convexity condition.
Keywords
- constructive mathematics, convex functions, fan theorem
ASJC Scopus subject areas
- Arts and Humanities(all)
- Philosophy
- Mathematics(all)
- Logic
Cite this
- Standard
- Harvard
- Apa
- Vancouver
- BibTeX
- RIS
In: Journal of Symbolic Logic, Vol. 83, No. 4, 12.2018, p. 1363-1375.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Brouwer's fan theorem and convexity
AU - Berger, Josef
AU - Svindland, G.
N1 - Publisher Copyright: © The Association for Symbolic Logic 2018.
PY - 2018/12
Y1 - 2018/12
N2 - In the framework of Bishop's constructive mathematics we introduce co-convexity as a property of subsets B of , the set of finite binary sequences, and prove that co-convex bars are uniform. Moreover, we establish a canonical correspondence between detachable subsets B of and uniformly continuous functions f defined on the unit interval such that B is a bar if and only if the corresponding function f is positive-valued, B is a uniform bar if and only if f has positive infimum, and B is co-convex if and only if f satisfies a weak convexity condition.
AB - In the framework of Bishop's constructive mathematics we introduce co-convexity as a property of subsets B of , the set of finite binary sequences, and prove that co-convex bars are uniform. Moreover, we establish a canonical correspondence between detachable subsets B of and uniformly continuous functions f defined on the unit interval such that B is a bar if and only if the corresponding function f is positive-valued, B is a uniform bar if and only if f has positive infimum, and B is co-convex if and only if f satisfies a weak convexity condition.
KW - constructive mathematics
KW - convex functions
KW - fan theorem
UR - http://www.scopus.com/inward/record.url?scp=85061910614&partnerID=8YFLogxK
U2 - 10.1017/jsl.2018.49
DO - 10.1017/jsl.2018.49
M3 - Article
VL - 83
SP - 1363
EP - 1375
JO - Journal of Symbolic Logic
JF - Journal of Symbolic Logic
SN - 0022-4812
IS - 4
ER -