When dealing with the problem of extending a Lipschitz function $f:A \to Y$ between metric spaces across an inclusion $A \hookrightarrow X$, one often imposes (conditions which imply) the following property on the target space $Y$. I'd like to know if this property has a name.
Let me describe our property: there exists a uniform bound $d \in (0,\infty)$ so that the following holds. Given (a) any finite subset $\{y_1,\ldots,y_n\} \subset Y$ of diameter smaller than $d$, and (b) any sequence $\{r_1,\ldots,r_n\}$ of strictly positive real numbers, define $$I(t) = \bigcap_1^n \text{Ball}(\text{center}=y_i,\text{radius}=r_it) $$ and let $\bar{t}$ be the infimum over all $t \geq 0$ so that $I(t) \neq \varnothing$. Then, our property says that the set $I(\bar{t})$ contains a unique element $y \in Y$.
See for instance the crucial Prop 4.2 in Lang/Schroeder's amazing 1997 GAFA paper here, which shows that every complete $Cat(k)$ space satisfies something stronger.
For what it's worth, I've been sorely tempted to call such metric spaces "Hellyscapes" after Helly's theorem, so you'll probably be doing the entire community a service by talking me out of it...
