A central problem in metric fixed point theory is to understand when a nonexpansive (i.e. Lipschitz with constant 1) self-map of a metric space has a fixed point. Even in the case where the metric space is a finite dimensional normed space, this is a subtle problem, as the map need not be a Lipschitz contraction or the space need not be bounded, so neither the contraction mapping theorem nor the Brouwer fixed point theorem applies. In this talk I will present necessary and sufficient conditions for a nonexpansive map on a finite dimensional normed space to have a bounded non-empty fixed point set. We will see how horofunctions and a famous problem in discrete geometry (the illumination problem) play a role in this problem. We will also discuss some applications to nonlinear eigenvalue problems arising in game theory and mathematical biology.
The talk involves basic mathematical analysis and metric geometry. It is very accessible and based on joint work with Brian Lins and Roger Nussbaum.