On the Lipschitz classification of normed spaces, unit balls, and spheres (Q2718961)

For every normed space \(E\), denote its closed unit ball and unit sphere by \(B_E\) and \(S_E\), respectively, and let the norm on the space \(E\oplus R\) be the sup norm. If \(F:(M, d)\to (M_0, d_0)\) (a function between metric spaces) is a bijection such that \(f\) and \(f^{-1}\) are \(k\)-Lipschitz functions, then we write \(M\sim M_0\). In the present paper the author proves the following theorems:NEWLINENEWLINENEWLINELet \(X\) and \(Y\) be normed spaces.NEWLINENEWLINENEWLINE(a) If \(B_X\sim B_Y\), then \(S_{X\oplus R}\sim S_{Y\oplus R}\);NEWLINENEWLINENEWLINE(b) If \(X\sim Y\), then \(S_{X\oplus R}\sim S_{Y\oplus R}\).NEWLINENEWLINENEWLINEThe following problem is open: Let \(X\) and \(Y\) be normed spaces such that \(S_{X\oplus R}\sim S_{Y\oplus R}\). Does it follow that \(S_X\sim S_Y\)?