The isometric extension problem between unit spheres of two separable Banach spaces (Q272507)

The author studies the following isometric extension problem proposed in [\textit{D.~Tingley}, Geom. Dedicata 22, 371--378 (1987; Zbl 0615.51005)]: Let \(X\) and \(Y\) be real Banach spaces with unit spheres \(S_X\) and \(S_Y\). If \(V_0: S_X \to S_Y\) is a surjective isometry, does \(V_0\) extend to a linear or affine isometry on \(X\)? As pointed out in the paper the answer is positive in many cases, e.g. for Hilbert spaces, \(\ell_p\) spaces, James space, and Tsirelson space to mention a few. The main result of this paper says that for a separable Banach space \(X\) the isometric extension problem of Tingley [loc. cit.] has a positive answer (in fact \(V_0\) extends to a linear isometry on \(X\)) provided the following condition holds: (*) For every \(x \in \mathrm{sm}(B_X)\) (the set of smooth points of the unit ball of \(X\)) there exists a positively increasing sequence \((r^{(0)}_n)_{n=1}^\infty\) with \(r^{(0)}_n \to \infty\) such that for every \(z \in\mathrm{sm}(B_X)\) the mapping \(V_0\) satisfies \(\|V_0z \pm r^{(0)}_nV_0x\| \leq \|z \pm r^{(0)}_nx\|\) for all natural numbers \(n\). The same conclusion is also proved to hold under (*) for reflexive spaces \(X\) and \(Y\).