# Fixed points in convex cones

## TL;DR

This paper introduces a fixed-point property for group actions on cones in topological vector spaces, extending classical theorems and establishing new fixed-point results for various classes of actions and groups.

## Contribution

It defines a new fixed-point property for group actions on cones, extending classical fixed-point theorems to broader contexts and establishing equivalences and closure properties.

## Key findings

- The fixed-point property always holds for equicontinuous actions.
- The property holds for distal actions on locally compact cones in the weak topology.
- The fixed-point property defines a group property stronger than amenability.

## Abstract

We propose a fixed-point property for group actions on cones in topological vector spaces. In the special case of equicontinuous actions, we prove that this property always holds; this statement extends the classical Ryll-Nardzewski theorem for Banach spaces. When restricting to cones that are locally compact in the weak topology, we prove that the property holds for all distal actions, thus extending the general Ryll-Nardzewski theorem for all locally convex spaces. Returning to arbitrary actions, the proposed fixed-point property becomes a group property, considerably stronger than amenability. Equivalent formulations are established and a number of closure properties are proved for the class of groups with the fixed-point property for cones.