Dear Luisa Malaguti

In general, the answer is negative by the following indirect argument. Assume X is separable and uniformly convex, and let G: X/K -->X be the map in question. If this map is Lipschitz, then it is a Lipschitz embedding, because the inverse map G^{-1} : G(X) \to X/K is also Lipschitz (in fact, it is a contraction) . This means, that X/K is Lipschitz embeddable into X. According to Theorem 3.5 of the paper

Heinrich, Stefan; Mankiewicz, P. Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces. Stud. Math. 73, 225-251 (1982),

https://eudml.org/doc/218444

this implies that X/K is isomorphic to a subspace of X. But the general result "each quotient space of a uniformly convex separable Banach space X is isomorphic to a subspace of X" is not correct. Say, L_5[0,1] has a quotient space isomorphic to l_4, but does not have subspaces isomorphic l_4.

This latter fact can be extracted from two old papers:

Kadets, M. I. Linear dimension of the spaces \(L_{p}\) and \(l_{q}\).

Usp. Mat. Nauk 13, No. 6(84), 95-98 (1958)

Kadets, M. I.; Pełczyński, Aleksander

Bases, lacunary sequences and complemented subspaces in the spaces \(L_ p\).

Stud. Math. 21, 161-176 (1962).

almost surely, the results are contained in some books like Lindenstrauss-Tzafriri Classical Banach spaces I and II.

All the best, Vladimir

Dear Prof. Kadets,

many thanks for your prompt and detailed answer.

I can understand your reasoning and it is really a pity that I have not Lipschitzianity in that context.

Cordial greetings, Luisa

Dear Prof. Luisa Malaguti ,

Although there is no Lipschitz condition, there is a good chance to demonstrate some uniform estimates using the modulus of convexity of the space.

Are such kind of estimates of interest for you?

Best, Vladimir