Let $X$ and $Y$ be two complete proper length spaces, $x \in X$ and $y \in Y$. Assume for every $r>0$ the closed balls $\overline{B_r(x)}$ and $\overline{B_r(y)}$ are isometric.

Does there exist an isometry $X \to Y$?

**Remark:** If you add that the isometries between the balls all map $x$ to $y$, you get a pointed isometry between $(X,x)$ and $(Y,y)$.

**Definitions:** A metric space is called *proper* if all closed balls are compact. A metric space is called *length space* if the distance of two points equals the infimum of the length of curves connecting these points, i.e. $d(x,y) = \inf \{ L(c) \mid c(0) = x, c(1) = y\}$.

In fact, for a *complete length space*, this infimum is a minimum.