### Abstract

We say that a complete metric space X has the fixed point property if every group of isometric automorphisms of X with a bounded orbit has a fixed point in X. We prove that if X is uniformly convex then the family of admissible subsets of X possesses uniformly normal structure and if so then it has the fixed point property. We also show that from other weaker assumptions than uniform convexity, the fixed point property follows. Our formulation of uniform convexity and its generalization can be applied not only to geodesic metric spaces.

Original language | English |
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Journal | Topology and its Applications |

DOIs | |

Publication status | Accepted/In press - 2014 Jan 31 |

### Fingerprint

### Keywords

- Bounded orbit
- Circumcenter
- Fixed point property
- Isometric action
- Normal structure
- Primary
- Secondary
- Uniform convexity

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

**Uniform convexity, normal structure and the fixed point property of metric spaces.** / Matsuzaki, Katsuhiko.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Uniform convexity, normal structure and the fixed point property of metric spaces

AU - Matsuzaki, Katsuhiko

PY - 2014/1/31

Y1 - 2014/1/31

N2 - We say that a complete metric space X has the fixed point property if every group of isometric automorphisms of X with a bounded orbit has a fixed point in X. We prove that if X is uniformly convex then the family of admissible subsets of X possesses uniformly normal structure and if so then it has the fixed point property. We also show that from other weaker assumptions than uniform convexity, the fixed point property follows. Our formulation of uniform convexity and its generalization can be applied not only to geodesic metric spaces.

AB - We say that a complete metric space X has the fixed point property if every group of isometric automorphisms of X with a bounded orbit has a fixed point in X. We prove that if X is uniformly convex then the family of admissible subsets of X possesses uniformly normal structure and if so then it has the fixed point property. We also show that from other weaker assumptions than uniform convexity, the fixed point property follows. Our formulation of uniform convexity and its generalization can be applied not only to geodesic metric spaces.

KW - Bounded orbit

KW - Circumcenter

KW - Fixed point property

KW - Isometric action

KW - Normal structure

KW - Primary

KW - Secondary

KW - Uniform convexity

UR - http://www.scopus.com/inward/record.url?scp=84930038586&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84930038586&partnerID=8YFLogxK

U2 - 10.1016/j.topol.2015.05.039

DO - 10.1016/j.topol.2015.05.039

M3 - Article

AN - SCOPUS:84930038586

JO - Topology and its Applications

JF - Topology and its Applications

SN - 0166-8641

ER -