Retraction from a unit ball onto its spherical cup (Q2821639)

Let \((H,\langle\cdot,\cdot\rangle)\) be a real Hilbert space, \(B\) denote the unit ball centered at the origin, \(S\) its unit sphere, and, let \(e\in S\). For each \(t\in [-1,1]\), let \(S_t= \{x\in B:\langle x,e\rangle\geq t\}\cap S\) and NEWLINE\[NEWLINE\kappa(t)= \text{inf}\{k:\text{there exists a \(k\)-Lipschitzian retraction from \(B\) onto }S_t\}.NEWLINE\]NEWLINE In the present paper, the authors give a new upper bound of \(\kappa(t)\) that simultaneously yields the precise formula of \(\kappa(t)\) for a finite-dimensional Hilbert space and a sharper upper bound of \(\kappa(t)\) for an infinite-dimensional Hilbert space.