An extension of a result of Ribe
If \(1<q<\infty\), \(1<p_ n<\infty\) and \(p_ n\to 1\) then \textit{M. Ribe} [Isr. J. Math. 48, 139-147 (1984; Zbl 0577.46015)] proved that the Banach spaces \(X=(\Sigma \oplus L_{p_ n})_ q\) and \(X\oplus L_ 1\) are uniformly homeomorphic but not Lipschitz equivalent. One of the results of the present paper shows that if \(1\leq p<\infty\), \(1\leq p_ n<\infty\), \(p_ n\to p\), \(1\leq q<\infty\), then \(X=(\Sigma \oplus \ell_{p_ n})_ q\) is uniformly homeomorphic to \(Y=X\oplus \ell_ p\), while if \(p_ n\neq p\), \(q\neq p\), Y is not Lipschitz homeomorphic to any subset of X.
uniformly homeomorphic but not Lipschitz equivalent
scientific article; zbMATH DE number 3992169