Retracting a ball in ℓ_1 onto its simple spherical cap - Intrakul, Jumpot; Iampiboonvatana, Smith Yokpaisan (Topological Methods in Nonlinear Analysis, 2024)
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 $κ(α)$, and the answer of the optimal retraction problem for the space $\ell_1$, denoted by $k_0(\ell_1)$.To be precise, there exists $-1< μ< 0$ such that $k_0(\ell_1)≤κ(α)≤2+k_0(\ell_1)$ whenever $-1< α< μ$; here $α$ is the level of spherical cap.