Take a piecewise linear curve $L$ in Euclidian space, i.e. a an ordered set of points $P$ sequentially connected by straight lines $l_{i}$, each defined by two points $p_i$ and $ p_{i+1}$.
Some such lines $L$ have the property that for any point not belonging to $L$, a closest point $p_c$ in $P$ will be one of the points defining a closest line $l_c$.
Is there a name for this property?
What conditions are sufficient?
One obvious case is when L itself is straight. My intuition suggests that bounding the sum of all angle differences between subsequent $l_i$ by $\pm90 ^{\circ}$ is a sufficient condition. Some joint condition on length and curvature should also be possible.
That $L$ does not cross itself seems to be a necessary condition.