# Lipschitz right inverses of Banach space quotients

Let $X$ be a Banach space and $Y$ a closed subspace of $X$. I am interested in quotients $q:X\to X/Y$ that do not have Lipschitz right inverses (not necessarily linear).

Of course, if $Y$ is complemented, then the quotient always has a Lipschitz (bounded linear) right inverse.

The only examples I know of that do not have Lipschitz right inverses are:

1. $\ell^\infty / c_0$ by Kalton [1, Theorem 4.2].
2. A quotient of a certain Càdlàg type space with $C[0,1]$ by Lindenstrauss and Aharoni [2, Remark ii].

I have two questions:

1. Are there any more examples known?
2. Is this perhaps a general feature of non-complemented subspaces of Banach spaces? Or is there an example of a non-complemented subspace $Y$ of some $X$ so that the quotient $q:X\to X/Y$ has a Lipschitz right inverse?

[1] Nigel Kalton. Lipschitz and uniform embeddings into ℓ∞. Fund. Math. 212 (2011), no. 1, 53–69.

[2] Aharoni, Israel; Lindenstrauss, Joram. Uniform equivalence between Banach spaces. Bull. Amer. Math. Soc. 84 (1978), no. 2, 281–283.

• This might be a naive approach, but assume that $X$ is uniformly convex (or even $X = L^p$ for $p \in (1,\infty)$) and let $\varphi: X/Y \ni x+Y \mapsto \varphi(x+Y) \in X$ select the proximum of $0$ in $x+Y$. Is $\varphi$ Lipschitz continuous? Jun 4 '18 at 19:51
• Additional remark: It is not difficult to show that the mapping $\varphi$ from my previous comment is Lipschitz continuous if its restriction to the unit sphere of $X/Y$ is Lipschitz continuous. But still... Jun 4 '18 at 19:53