# A question on metric characterization of approximation property

My question is: a Banach space $X$ has the approximation property if and only if for every $\epsilon>0$ and every compact subset $K\subset X$, there exists a Lipschitz map $T: X\rightarrow X$ with relatively compact image $T(X)$ such that $\|Tx-x\|<\epsilon$, for all $x\in K$. The purpose of my question is to know about metric characterizations of approximation property. Is there any reference on metric characterizations of approximation property?

• I don't think anything is known other than Theorem 5.3 in Godefroy, G.; Kalton, N. J. Lipschitz-free Banach spaces. Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday. Studia Math. 159 (2003), no. 1, 121–141. – Bill Johnson Feb 13 '15 at 23:04
• Do you think the proof of Theorem 5.3 in Godefroy and Kalton's paper works for the AP? – Dongyang Chen Feb 14 '15 at 14:14
• Do you think my question is interesting or not? – Dongyang Chen Feb 14 '15 at 16:13
• Sure, it is interesting. – Bill Johnson Feb 15 '15 at 15:02