# Selection type results and fixed point property for affine bi-Lipschitz maps

## TL;DR

This paper refines a selection principle in Banach spaces and demonstrates the existence of fixed point free affine bi-Lipschitz maps on certain convex subsets under weaker topologies.

## Contribution

It introduces a refined selection principle and applies it to show fixed point free affine bi-Lipschitz maps on non-weakly compact convex sets in Banach spaces.

## Key findings

- Refined selection principle for Banach space sequences.
- Existence of fixed point free affine bi-Lipschitz maps.
- Construction of weaker topologies with specific fixed point properties.

## Abstract

We obtain a refinement of a selection principle for $(\mathcal{K}, \lambda)$-wide-$(s)$ sequences in Banach spaces due to Rosenthal. This result is then used to show that if $C$ is a bounded, non-weakly compact, closed convex subset of a Banach space $X$, then there exists a Hausdorff vector topology $\tau$ on $X$ which is weaker than the weak topology, a closed, convex $\tau$-compact subset $K$ of $C$ and an affine bi-Lipschitz map $T: K\to K$ without fixed points.