This is an announcement for the paper “Stability constants of the weak$^*$ fixed point property for the space $\ell_1$” by Emanuele Casini<https://arxiv.org/find/math/1/au:+Casini_E/0/1/0/all/0/1>, Enrico Miglierina<https://arxiv.org/find/math/1/au:+Miglierina_E/0/1/0/all/0/1>, Łukasz Piasecki<https://arxiv.org/find/math/1/au:+Piasecki_L/0/1/0/all/0/1>, Roxana Popescu<https://arxiv.org/find/math/1/au:+Popescu_R/0/1/0/all/0/1>. Abstract: The main aim of the paper is to study some quantitative aspects of the stability of the weak∗ fixed point property for nonexpansive maps in $\ell_1$ (shortly, $w^*$-fpp). We focus on two complementary approaches to this topic. First, given a predual $X$ of $\ell_1$ such that the $\sigma(\ell_1, X)$-fpp holds, we precisely establish how far, with respect to the Banach-Mazur distance, we can move from X without losing the $w^*$-fpp. The interesting point to note here is that our estimate depends only on the smallest radius of the ball in $\ell_1$ containing all $\sigma(\ell_1, X)$-cluster points of the extreme points of the unit ball. Second, we pass to consider the stability of the $w^*$-fpp in the restricted framework of preduals of $\ell_1$. Namely, we show that every predual $X$ of $\ell_1$ with a distance from $c_0$ strictly less than 3, induces a weak∗ topology on $\ell_1$ such that the $\sigma(\ell_1, X)$-fpp holds. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.02133