# Nearly injective Banach spaces

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?).

• 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. – Mikhail Ostrovskii Jan 24 '16 at 19:08
• Nice! @MikhailOstrovskii -- thank you for quoting the H+M paper. – Włodzimierz Holsztyński Jan 25 '16 at 11:00