Retracting a ball in \(\ell_1\) onto its simple spherical cap (Q6545457)

Let \(X\) be a (topological) space and \(A\) a subspace of \(X\). A \textbf{retraction} of \(X\) onto \(A\) (making \(A\) a \textbf{retract} of \(X\)) is a continuous map \(r:X\rightarrow A\) satisfying \(r|_A=id_A\) (the identity map on \(A\)). Given metric spaces \(X\) and \(Y\), a map \(f:X\rightarrow Y\) is \textbf{Lipschitz} if there is a constant \(C>0\) (called a \textbf{Lipschitz constant}) such that \(d_Y(f(x),f(x'))\leq Cd_X(x,x')\), for all \(x,x'\in X\).\N\NIt is known that a Banach space \(X\) is infinite-dimensional if and only if there exists a Lipschitz retraction \(r:B_X\rightarrow S_X\), of the \textbf{closed unit ball} \(B_X\) (centered at \(0\)) onto the \textbf{unit sphere} \(S_X:=\partial B_X\) in \(X\), [\textit{B. Nowak}, Bull. Acad. Pol. Sci., Sér. Sci. Math. 27, 861--864 (1979; Zbl 0472.54008)] and [\textit{Y. Benyamini} and \textit{Y. Sternfeld}, Proc. Am. Math. Soc. 88, 439--445 (1983; Zbl 0518.46010)]. So, given an infinite-dimensional Banach space \(X\), the number \(k_0(X)\) denoting the \textbf{smallest Lipschitz constant} for such a retraction is of interest.\N\NMore generally, write any infinite-dimensional Banach space \(X\) in the form \(X=\mathbb{R} e\oplus X_e\), for a unit vector \(e\in X\). For \(-1\leq\alpha\leq 1\), define the \textbf{\((\alpha,e)\)-spherical cap} (or level \(\alpha\) spherical cap along \(e\in X\)) in \(X\) by \(S_{X,\alpha,e}:=\{te\oplus z\in X:t\geq\alpha,z\in X_e\}\cap S_X\). Then, whenever it exists or makes sense, let \(k(\alpha)\equiv k_{1+\alpha}(X)\) denote the \textbf{smallest Lipschitz constant} for retractions \(r:B_X\rightarrow S_{X,\alpha,e}\) (where \(S_{X,e,-1}=S_X\) and \(k(-1)=k_0(X)\)).\N\NThe problem of finding \(k_0(X)\) for an arbitrary infinite-dimensional Banach space \(X\) (called \textbf{optimal retraction problem}) is yet unsolved, with progress (according to this paper and [\textit{J. Intrakul} et al., Topol. Methods Nonlinear Anal. 52, No. 2, 677--691 (2018; Zbl 1486.47093)]) consisting only of approximations to a few special cases, including \(k_0(X)\geq 3\), \(k_0(\ell^1)\in[4,8]\), \(k_0(L^1[0,1])\in[3,8]\), \(k_0(BC_z(M))\in[3,2(2+\sqrt{2})]\), and \(k_0(\mathcal{H})\in(4.58, 28.99)\), where \(\ell^1\) is the Banach space of real sequences with norm \(\|(x_n)\|:=\sum_n|x_n|\), \(BC_z(M)\) is the space of bounded continuous real-valued functions on a connected metric space \(M\) consisting of more than one point and vanishing at \(z\) in \(M\), and \(\mathcal{H}\) is an infinite-dimensional Hilbert space.\N\NFor the Hilbert space \(\mathcal{H}\), the retractions \(r:B_\mathcal{H}\rightarrow S_{\mathcal{H},\alpha,e}\) always exist [\textit{P. Chaoha} et al., Topol. Methods Nonlinear Anal. 40, No. 1, 215--224 (2012; Zbl 1290.47054)] and, under suitable conditions, \(k(\alpha)=k_{1+\alpha}(\mathcal{H})\) can be used to find \(k_0(\mathcal{H})\) [\textit{J. Intrakul} et al., Topol. Methods Nonlinear Anal. 52, No. 2, 677--691 (2018; Zbl 1486.47093)]. Retractions of the ball onto spherical caps (with connections to \(k_0(\cdot)\)) have been studied for sequence spaces such as \(c_0,c,\ell^\infty\) [\textit{K. Goebel}, Ann. Univ. Mariae Curie-Skłodowska, Sect. A 74, No. 1, 45--55 (2020; Zbl 07269210)], where \(\ell^\infty\) is the Banach space of real sequences with norm \(\|(x_n)\|:=\sup_n|x_n|\), \(c\) the subspace of \(\ell^\infty\) consisting convergent sequences, and \(c_0\) the subspace of \(c\) consisting of sequences that converge to zero.\N\NInspired by the latter, this paper defines spherical caps in \(\ell^1\) and obtains estimates (via Theorem 3.1) for \(k(\alpha)=k_{1+\alpha}(\ell^1)\) in terms of bounds (upper and lower) that are functions of \(k_0(\ell^1)\), where \(k(\alpha)\) is exactly determined for \(\alpha\in\{-1\}\cup[0,1]\) by \(k(1)=0\), \(k(-1)=k_0(\ell^1)\), and \(k_0(\alpha)=2\) for \(\alpha\in[0,1)\).