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There was a problem about nearly injective metric spaces posed by Aronszajn and Panitchpakdi which I actually solved in the past but it still remains open (as long as I know) for the Banach spaces--so this is the QUESTION for this thread. I feel that the Banach version is still more interesting.

Term injective in the context of the metric spaces and of the Banach spaces is equivalent to the A&P's term hyperconvex.

Let $\ m\ $ be a cardinal number. A metric space $\ (X\ d)\ $ is said to be $m$-hyperconvex $\ \Leftarrow:\Rightarrow\ $ for every $\ A\subseteq X\ $ such that $\ |A|<m,\ $ and for each $\ f:A\rightarrow \mathbb R\ $ satisfying

$$\forall_{x\ y\in A}\ \ f(x)+f(y) \ge d(x\ y)$$

there exists $\ a\in A\ $ for which $\ d(a\ x)\le f(x)\ $ for all $\ x\in A$.

Thus, to answer Aronszajn & Panitchpakdi's question in positive, for each inequality $\ 4\le m <\aleph_0\ $ there exits an $m$-hyperconvex space which is not an $(m+1)$-hyperconvex (also, $\ m=2\ $ is trivial, and $\ m=3\ $ is a mood improver). But

are there $m$-hyperconvex Banach spaces which are not $(m+1)$-hyperconvex? (Can they be finitely dimensional?).

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    $\begingroup$ The following paper can be of interest in this context: Hansen, Lima, The structure of finite-dimensional Banach spaces with the 3.2. intersection property. Acta Math. 146 (1981), no. 1-2, 1-23. In this paper it is proved that any finite-dimensional Banach space $X$ with the 3.2.I.P. is obtained from the real line by repeated $\ell_1$ and $\ell_\infty$-summands. The 3.2.I.P. is the property that any 3 balls must intersect if they intersect in pairs. $\endgroup$
    – Mikhail Ostrovskii
    Commented Jan 24, 2016 at 19:08
  • $\begingroup$ Nice! @MikhailOstrovskii -- thank you for quoting the H+M paper. $\endgroup$
    – Włodzimierz Holsztyński
    Commented Jan 25, 2016 at 11:00

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