Abstract:
One of the most intriguing problems in metric fixed point theory is whether we can find closed convex and unbounded subsets of Banach spaces with the fixed point property. A celebrated theorem due to W.O. Ray in 1980 states that this cannot happen if the space is Hilbert. This problem was so poorly understood that two antagonistic questions were raised: the first one was if this property characterizes Hilbert spaces within the class of Banach spaces, while the second one asked if this property characterizes any space at all, that is, if Ray's theorem states in any Banach space. The second problem is still open but the first one has recently been answered in the negative by T. Domínguez Benavides after showing that Ray's theorem also holds true in the classical space of real sequences $c_0$.
The situation seems, however, to be completely different for CAT(0) spaces. Although Hilbert spaces are a particular class of CAT(0) spaces, there are different examples of CAT(0) spaces, including $\mathbb{R}$-trees, in the literature for which we can find closed convex and unbounded subsets with the fixed point property. In this talk we will look closely at this problem. First, we will introduce a geometrical condition inspired in the Banach-Steinhaus theorem for CAT(0) spaces under which we can still assure that Ray's theorem holds true. We will provide different examples of CAT(0) spaces with this condition but we will notice that all these examples are of a very strong Hilbertian nature. Then we will look at $\delta$-hyperbolic geodesic spaces. If looked from very far these spaces, if unbounded, resemble $\mathbb{R}$-trees, therefore it is natural to try to find convex closed and unbounded subsets with the fixed point property in these spaces. We will present some partial results in this direction.
This talk is based a joint work with Bożena Piątek.