"Lion-Man" and the Fixed Point Property

TL;DR

This paper explores the connection between fixed point properties of continuous maps and a lion and man game in convex spaces, establishing compactness as key to both fixed points and game success.

Contribution

It proves that in locally compact geodesic spaces, compactness, fixed point property, and lion success are equivalent, linking geometric rays to these properties.

Findings

Compactness is equivalent to fixed point property.

Success of the lion game characterizes compactness.

Existence of rays explains these equivalences.

Abstract

This paper focuses on the relation between the fixed point property for continuous mappings and a discrete lion and man game played in a strongly convex domain. Our main result states that in locally compact geodesic spaces, the compactness of the domain is equivalent to its fixed point property, as well as to the success of the lion. The common link among these properties involves the existence of different types of rays, which we also discuss.