Retracting a ball in $\ell_1$ onto its simple spherical cap

In this article, a notion and classification of spherical caps in the sequence space $\ell_1$ are introduced, and the least Lipschitz constant of Lipschitz retractions from the unit ball onto a spherical cap is defined. In addition, an approximation of this value for the specific spherical cap, the simple spherical cap, is calculated. This approximation reveals a rough relation between these values, denoted by $\kappa(\alpha)$, and the answer of the optimal retraction problem for the space $\ell_1$, denoted by $k_0(\ell_1)$. To be precise, there exists $-1< \mu< 0$ such that $k_0(\ell_1)\leq\kappa(\alpha)\leq2+k_0(\ell_1)$ whenever $-1< \alpha< \mu$; here $\alpha$ is the level of spherical cap.