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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.

Erik
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  • The common name for what you call a "piecewise straight line" is "piecewise linear curve". The endpoints of the line segments are called "vertices". Your condition appears to be of too little importance to have been given a well-known name, so you are free to name it as you choose. Just don't expect your name to become well-known either. I expect any convex piecewise linear curve to have this property. – Paul Sinclair Jun 16 '16 at 12:43
  • Thanks! Fixed the terminology. If I understand correctly, one counterexample to the convex piecewise linear curve condition is the curve (0,0)->(10,0)->(0,10)->(0,3), where the point (3,2) has (0,3) as closest vertex and (0,0)->(10,0) as closest edge. – Erik Jun 16 '16 at 14:11

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