This is an announcement for the paper “Komlós' Theorem and the Fixed Point Property for affine mappings” by T. Domínguez Benavideshttps://arxiv.org/find/math/1/au:+Benavides_T/0/1/0/all/0/1, M. Ahttps://arxiv.org/find/math/1/au:+A_M/0/1/0/all/0/1, Japónhttps://arxiv.org/find/math/1/au:+Jap%5C%27on/0/1/0/all/0/1.
Abstract: Assume that $X$ is a Banach space of measurable functions for which Koml'os' Theorem holds. We associate to any closed convex bounded subset $C$ of $X$ a coefficient $t(C)$which attains its minimum value when $C$ is closed for the topology of convergence in measure and we prove some fixed point results for affine Lipschitzian mappings, depending on the value of $t(C)\in {1, 2]$and the value of the Lipschitz constants of the iterates. As a first consequence, for every $L<2$, we deduce the existence of fixed points for affine uniformly $L$-Lipschitzian mappings defined on the closed unit ball of $L_1([0,1])$. Our main theorem also provides a wide collection of convex closed bounded sets in $L_1([0,1])$ and in some other spaces of functions, which satisfy the fixed point property for affine nonexpansive mappings. Furthermore, this property is still preserved by equivalent renormings when the Banach-Mazur distance is small enough. In particular, we prove that the failure of the fixed point property for affine nonexpansive mappings in $L_1(\mu)$ can only occur in the extremal case $t(C)=2$. Examples are displayed proving that our fixed point theorem is optimal in terms of the Lipschitz constants and the coefficient $t(C)$.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1709.03333