Isometries between unit spheres of the \(\ell^{\infty}\)-sum of strictly convex normed spaces (Q2872002)

Tingley's famous isometric extension problem asks whether any surjective isometry between unit spheres of real normed spaces \(E\) and \(F\) can be extended to a linear isometry between the whole spaces \(E\) and \(F\). The authors give an affirmative answer to this problem for the particular case of \(E =(\bigoplus_{\gamma \in \Gamma}E_\gamma)_{ \ell^\infty}\) and \(F=(\bigoplus_{\delta \in \Delta}F_\delta)_{\ell^\infty}\) being \(\ell^{\infty}\)-sums of two collections of strictly convex real normed spaces, where each set \(\Gamma\) and \(\Delta\) contains at least 2 elements. Recall that in general Tingley's problem remains open even for spaces of dimension 2.